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Rainbow Colouring of Split and Threshold Graphs
Abstract
A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chordal Graph is a graph in which every cycle of length more than 3 has a chord. A Split Graph is a chordal graph whose vertices can be partitioned into a clique and an independent set. A threshold graph is a split graph in which the neighbourhoods of the independent set vertices form a linear order under set inclusion. In this article, we show the following:
1
The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum.
2
For every integer k ≥ 3, the problem of deciding whether a graph can be rainbow coloured using k colours remains NP-complete even when restricted to the class of chordal graphs.
3
For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Further, we can optimally rainbow colour a threshold graph in linear time.
... Given a graph G and any fixed k ≥ 2, determining if rc(G) ≤ k is NP-complete [1,3,8], which implies that determining rc(G) in general is NP-hard [3]. In fact, the NP-hardness remains even when restricted to bipartite graphs for any fixed k ≥ 3 [8], to chordal graphs for any fixed k ≥ 2 [4], and to split graphs for fixed k ∈ {2, 3} [4]. In view of this hardness, several authors engaged efforts in the searching for upper and lower bounds for the rainbow connection number [3,4,18,19]. ...
... Given a graph G and any fixed k ≥ 2, determining if rc(G) ≤ k is NP-complete [1,3,8], which implies that determining rc(G) in general is NP-hard [3]. In fact, the NP-hardness remains even when restricted to bipartite graphs for any fixed k ≥ 3 [8], to chordal graphs for any fixed k ≥ 2 [4], and to split graphs for fixed k ∈ {2, 3} [4]. In view of this hardness, several authors engaged efforts in the searching for upper and lower bounds for the rainbow connection number [3,4,18,19]. ...
... In fact, the NP-hardness remains even when restricted to bipartite graphs for any fixed k ≥ 3 [8], to chordal graphs for any fixed k ≥ 2 [4], and to split graphs for fixed k ∈ {2, 3} [4]. In view of this hardness, several authors engaged efforts in the searching for upper and lower bounds for the rainbow connection number [3,4,18,19]. For instance, an immediate lower bound is the diameter of G, i.e. the largest distance between a ✩ Partial results concerning some topics in this full paper have been presented at local events and published in the corresponding annals (Rocha and Almeida, 2017 [15,16]; Rocha and Almeida, 2019 [17]). ...
Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention in the last years in Combinatorics. In particular, the rainbow connection number of a connected graph G, denoted rc(G), is the least k for which G admits a (not necessarily proper) k-edge-coloring such that between any pair of vertices there is a path whose edge colors are all distinct. When G is disconnected, we define rc(G)=∞. A graph G is rainbow critical if the deletion of any edge of G increases rc(G). In this paper we determine the rainbow connection number for the shadow graphs of paths, some circulant graphs, and some Cartesian products involving cycles and paths. We also determine conditions for the rainbow criticality and non-criticality of wheels, fans, and some Cartesian products involving cycles, paths, pans, and complete graphs.
... Chakraborty, Fischer, Matsliah, and Yuster [6] investigated the hardness and algorithms for the rainbow connection number, and showed that given a graph G, deciding if rc(G) = 2 is NP-complete. For further algorithmic results, we refer the interest reader to [1,7,9]. ...
... Furthermore, let P 1 be the subpath between x and u on P , let P 2 be the subpath between y and v on P . By the definition of edge-coloring c, the paths P 1 and P 2 are rainbow paths of (G ∪ H, c) such that c( [7] Chandran, and Rajendraprasad proved that rc(T ) = n − 1 for a tree T of order n. Let T 1 and T 2 be two nontrivial trees such that |V (T 1 ) ∩ V (T 1 )| = 1. ...
A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. An edge-colored graph G is rainbow connected if every two distinct vertices are connected by a rainbow path. The rainbow connection number rc(G) of G is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we study bounds of rainbow connection number of some graph operations, such as the union of two graphs, adding edges, deleting edges, and adding vertices and edges. Moreover, we also study the following extremal problem. Let k and n be two integers such that 1≤k≤ℓ<n. Find the smallest integer f(n,k,ℓ) such that for each graph G of order n and diameter k, there exists an edge set F⊆E(G-) satisfying |F|≤f(n,k,ℓ) and rc(G+F)≤ℓ.
... It may be worth noting that Spl m (K n ) is a split graph, a graph whose vertex-set can be partitioned into an independent set and a clique. A nearly optimal algorithm has been announced [3] that gives a rainbow coloring of any split graph G using at most rc(G) + 1 colors in linear time. However, it is unlikely that any polynomial time optimal algorithm can be found, because deciding rc(G) ≤ 3 is already NP-hard even when G is restricted to be in the class of split graphs [3] . ...
... A nearly optimal algorithm has been announced [3] that gives a rainbow coloring of any split graph G using at most rc(G) + 1 colors in linear time. However, it is unlikely that any polynomial time optimal algorithm can be found, because deciding rc(G) ≤ 3 is already NP-hard even when G is restricted to be in the class of split graphs [3] . ...
An edge-coloring of a graph is called rainbow if any two vertices are connected by a path consisting of edges of different colors. The least number of colors in such a coloring is called the rainbow connection number of G, denoted by rc(G). An edge-coloring of a graph is called strong rainbow if any two vertices are connected by a geodesic consisting of edges of different colors. The least number of colors in such a coloring is called the strong rainbow connection number of G, denoted by src(G). In this paper we study the rc and src of the m-splitting of a graph. In particular we study Splm(Kn). We present the exact values of its rc and src in several cases, and we prove several bounds in the other cases.
... Chandran et al. [10] showed that for split graphs, the problem of deciding if rc(G) = k is NP-complete for k ∈ {2, 3}, and in P for all other values of k. Chandran and Rajendraprasad [11] showed split graphs can be rainbow colored in linear time using at most one more color than the optimum. In the same paper, the authors also give a linear time algorithm for finding a minimum rainbow coloring of a threshold graph. ...
... Equivalently, a graph is chordal if it contains no induced cycle of length 4 or more. The problem of deciding whether rc(G) = k, for every integer k ≥ 3 remains NP-complete for the class of chordal graphs [11]. It follows from [10] that deciding if a chordal graph can be strong rainbow colored using k colors is NP-complete for k = 2. ...
A path in an edge-colored graph $G$ is rainbow if no two edges of it are
colored the same. The graph $G$ is rainbow colored if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph $G$ is strong rainbow colored. The minimum
number of colors needed to make $G$ rainbow colored is known as the rainbow
connection number, and is denoted by $\text{rc}(G)$. The minimum number of
colors needed to make $G$ strong rainbow colored is known as the strong rainbow
connection number, and is denoted by $\text{src}(G)$. A graph is chordal if it
contains no induced cycle of length 4 or more. We consider the rainbow and
strong rainbow connection numbers of block graphs, which form a subclass of
chordal graphs. We give an exact linear time algorithm for strong rainbow
coloring block graphs exploiting a clique tree representation each chordal
graph has. In contrast, we prove that for every integer $k \geq 3$, it is $\sf
NP$-complete to decide if a given block graph or a bridgeless block graph can
be rainbow colored with at most $k$ colors. We derive a tight upper bound of
$|S|+2$ on $\text{rc}(G)$, where $G$ is a block graph, and $S$ its set of
minimal separators. We also present an algorithm for rainbow coloring a block
graph $G$ with $b$ bridges using at most $3/2 \text{rc}(G) + b + 1$ colors.
... Given a graph G, determining if rc(G) ≤ k, for any fixed k ≥ 2, is an NP-complete problem [Chakraborty et al. 2011, Ananth et al. 2011, Li and Li 2011. In view of this hardness, several authors engaged efforts in the searching for upper and lower bounds for the rainbow connection number [Chakraborty et al. 2011, Chandran and Rajendraprasad 2012, Schiermeyer 2009]. Despite such effort, the problem is completely solved only for specific and structured graph classes, such as complete graphs, paths, and cycles [Chartrand et al. 2008]. ...
Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention last years in Combinatorics. The rainbow connection number of a graph G is the least number of colors for a (not necessarily proper) edge coloring of G such that between any pair of vertices there is a path whose edge colors are all distinct. In this paper we determine the rainbow connection number of the triple triangular snake graphs.
... [Chakraborty et al. 2011, Ananth et al. 2011, Li and Li 2011. Considerando a dificuldade em se calcular o ( ), diversos autores empreenderam esforços na busca por limitantes inferiores e superiores para o número de conexão arcoíris [Chakraborty et al. 2011, Chandran and Rajendraprasad 2012, Schiermeyer 2009]. ...
A rainbow coloring of a connected graph 𝐺 is an edge coloring that is not necessarily proper such that there is a path between any pair of vertices of 𝐺 whose edge colors are pairwise distinct. The rainbow connection number of a graph 𝐺, denoted by 𝑟𝑐(𝐺), is the least number of colors for which there is a rainbow coloring of 𝐺. A graph 𝐺 is rainbow critical if its rainbow connection number increases when we remove any edge from 𝐺. In this work, we prove wheel and fan graphs are rainbow critical only when these graphs
have few vertices.
... An alternative hardness proof for every k > 1 was provided by Le and Tuza [18]. For complexity results on restricted graph classes, see e.g., [4, 5, 6, 12]. algorithm running in time 2 |S| n O(1) , which is tight up to a polynomial factor. ...
The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in $k$ colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any $k\ge 2$, there is no algorithm for Rainbow k-Coloring running in time $2^{o(n^{3/2})}$, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set $S$ of pairs of vertices and we ask if there is a coloring in which all the pairs in $S$ are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by $|S|$. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer $q$ and we ask if there is a coloring in which at least $q$ anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by $q$ and has a kernel of size $O(q)$ for every $k\ge 2$ (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011].
... The Cayley graph Γ(G, S) on G with connection set S is a graph that has as its vertices the elements of G and is such that it has an edge e joining vertices g and h if and only if h = g + s, for some s ∈ S. In this case, we say that the edge e has color s. A concept of "rainbow" has been used in various fashions in a graph theory context, in [1,2,3,8,9,10,14,15,16,17,18,19,20,21] and related papers. Ours is in relation to edge colors in Cayley graphs of finite cyclic groups. ...
Let Tn be the complete undirected Cayley graph of the odd cyclic group Zn. Connected graphs whose vertices are rainbow tetrahedra in Tn are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs G of largest degree 6, asymptotic diameter |V (G)|1/3 and almost all vertices with degree: (a) 6 in G; (b) 4 in exactly six connected subgraphs of the (3, 6, 3, 6)-semi- regular tessellation; and (c) 3 in exactly four connected subgraphs of the {6, 3}-regular hexagonal tessellation. These vertices have as closed neigh- borhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations.
... The Cayley graph Γ(G, S) on G with connection set S is a graph that has as its vertices the elements of G and is such that it has an edge e joining vertices g and h if and only if h = g + s, for some s ∈ S. In this case, we say that the edge e has color s. A concept of "rainbow" has been used in various fashions in a graph theory context, in [1,2,3,8,9,10,14,15,16,17,18,19,20,21] and related papers. Ours is in relation to edge colors in Cayley graphs of finite cyclic groups. ...
Arising from complete Cayley graphs $\Gamma_n$ of odd cyclic groups $\Z_n$,
an asymptotic approach is presented on connected labeled graphs whose vertices
are labeled via equally-multicolored copies of $K_4$ in $\Gamma_n$ with
adjacency of any two such vertices whenever they are represented by copies of
$K_4$ in $\Gamma_n$ sharing two equally-multicolored triangles. In fact, these
connected labeled graphs are shown to form a family of graphs of largest degree
6 and diameter asymptotically of order $|V|^{1/3}$, properties shared by the
initial member of a collection of families of Cayley graphs of degree $2m\geq
6$ with diameter asymptotically of order $|V|^{1/m}$, where $3\leq m\in\Z$.
For a graph G and cR:E(G)→[k], a path P between u,v∈V(G) is a rainbow path if for distinct e,e′∈E(P), we have cR(e)≠cR(e′). Rainbow k-Coloring takes a graph G and the objective is to check if there is cR:E(G)→[k] such that for all u,v∈V(G) there is a rainbow path between u and v. Two variants of the above problem are Subset Rainbow k-Coloring and Steiner Rainbow k-Coloring, where we are additionally given a subset S⊆V(G)×V(G) and S′⊆V(G), respectively. Moreover, the objective is to check if there is cR:E(G)→[k], such that there is a rainbow path for each (u,v)∈S and u,v∈S′, respectively. Under ETH, we obtain that for each k≥3:
1.Rainbow k-Coloring has no 2o(|E(G)|)nO(1)-time algorithm.
2.Steiner Rainbow k-Coloring has no 2o(|S|2)nO(1)-time algorithm.
We also obtain that Subset Rainbow k-Coloring and Steiner Rainbow k-Coloring admit 2O(|S|)nO(1)- and 2O(|S|2)nO(1)-time algorithms, respectively.
A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (src(G)) of the graph. When proving upper bounds on src(G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number (vsrc(G)). In this paper, we give upper bounds on vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on src(G) for these classes, showing that the study of vsrc(G) enables meaningful progress on bounding src(G). Then we study the complexity of the problem to compute vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for src(G). We also observe that deciding whether vsrc(G)=k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether vsrc(G)≤3 nor to approximate vsrc(G) within a factor n1-ε, unless P=NP.
An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by \({{\mathrm{rc}}}(G)\), is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the Strong Rainbow Vertex Coloring problem is \(\textsf {NP}\)-complete even on graphs of diameter 3. We show that when the number of colors is fixed, then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.
An edge-colored graph $G$ is said to be rainbow connected if between each
pair of vertices there exists a path which uses each color at most once. The
rainbow connection number, denoted by $rc(G)$, is the minimum number of colors
needed to make $G$ rainbow connected. Along with its variants, which consider
vertex colorings and/or so-called strong colorings, the rainbow connection
number has been studied from both the algorithmic and graph-theoretic points of
view.
In this paper we present a range of new results on the computational
complexity of computing the four major variants of the rainbow connection
number. In particular, we prove that the \textsc{Strong Rainbow Vertex
Coloring} problem is $NP$-complete even on graphs of diameter $3$. We show that
when the number of colors is fixed, then all of the considered problems can be
solved in linear time on graphs of bounded treewidth. Moreover, we provide a
linear-time algorithm which decides whether it is possible to obtain a rainbow
coloring by saving a fixed number of colors from a trivial upper bound.
Finally, we give a linear-time algorithm for computing the exact rainbow
connection numbers for three variants of the problem on graphs of bounded
vertex cover number.
After a necessary condition is given, 3-rainbow coloring of split graphs with time complexity O(m) is obtained by constructive method. The number of corresponding colors is at most 2 or 3 more than the minimum number of colors needed in a 3-rainbow coloring.
A rainbow path in an edge coloured graph is a path in which no two edges are
coloured the same. A rainbow colouring of a connected graph G is a colouring of
the edges of G such that every pair of vertices in G is connected by at least
one rainbow path. The minimum number of colours required to rainbow colour G is
called its rainbow connection number. Between them, Chakraborty et al. [J.
Comb. Optim., 2011] and Ananth et al. [FSTTCS, 2012] have shown that for every
integer k, k \geq 2, it is NP-complete to decide whether a given graph can be
rainbow coloured using k colours.
A split graph is a graph whose vertex set can be partitioned into a clique
and an independent set. Chandran and Rajendraprasad have shown that the problem
of deciding whether a given split graph G can be rainbow coloured using 3
colours is NP-complete and further have described a linear time algorithm to
rainbow colour any split graph using at most one colour more than the optimum
[COCOON, 2012]. In this article, we settle the computational complexity of the
problem on split graphs and thereby discover an interesting dichotomy.
Specifically, we show that the problem of deciding whether a given split graph
can be rainbow coloured using k colours is NP-complete for k \in {2,3}, but can
be solved in polynomial time for all other values of k.
The concept of rainbow connection was introduced by Chartrand et al. in 2008.
It is fairly interesting and recently quite a lot papers have been published
about it. In this survey we attempt to bring together most of the results and
papers that dealt with it. We begin with an introduction, and then try to
organize the work into five categories, including (strong) rainbow connection
number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection
number, algorithms and computational complexity. This survey also contains some
conjectures, open problems or questions.
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. We prove that p=lnn/m is a sharp threshold function for the property rc(G(n,m,p))≤3 in random bipartite graph model G(n,m,p).
The concept of rainbow connection was introduced by Chartrand et al. in 2008.
It is fairly interesting and recently quite a lot papers have been published
about it. In this survey we attempt to bring together most of the results and
papers that dealt with it. We begin with an introduction, and then try to
organize the work into five categories, including (strong) rainbow connection
number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection
number, algorithms and computational complexity. This survey also contains some
conjectures, open problems or questions.
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In the first result of this paper we prove that computing rc(G) is NP-Hard solving an open problem from Caro et al. (Electron. J. Comb. 15, 2008, Paper R57). In fact, we prove that it is already NP-Complete to decide if rc(G)=2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ε>0, a connected graph with minimum degree at least ε
n has bounded rainbow connection, where the bound depends only on ε, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.
An edge colored graph G is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected.
In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold \(p=\frac{\log n+\omega }{n}\) where ω = ω(n) → ∞ and ω = o(logn) and of random r-regular graphs where r ≥ 3 is a fixed integer. Specifically, we prove that the rainbow connectivity rc(G) of G = G(n,p) satisfies \(rc(G) \sim \max\left\{Z_1,diameter(G)\right\}\) with high probability (whp). Here Z
1 is the number of vertices in G whose degree equals 1 and the diameter of G is asymptotically equal to \(\frac{ \log{n}}{\log{\log{n}}}\)
whp. Finally, we prove that the rainbow connectivity rc(G) of the random r-regular graph G = G(n,r) satisfies rc(G) = O(log2n) whp.
A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by ( src (G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results can be summarised as below: 1) For every fixed k≥3, it is NP-complete to decide whether src (G)≤k even when the graph G is bipartite. 2) For every fixed odd k≥3, it is NP-complete to decide whether rc (G)≤k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. 3) The following problem is fixed parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. 4) For a directed graph G, it is NP-complete to decide whether rc (G)≤2.
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that
rc(G) n4rc(G) for graphs with minimum degree three, which was conjectured by Caro et al. [Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R.
Yuster, On rainbow connection, The Electronic Journal of Combinatorics 15 (2008), #57.]
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we prove several non-trivial upper bounds for rc(G), as well as determine sucient conditions that guarantee rc(G) = 2. Among our results we prove that if G is a connected graph with n vertices and with minimum degree 3 then rc(G) < 5n/6, and if the minimum degree is then rc(G) ln n(1 + o (1)). We also determine the threshold function for a random graph to have rc(G) = 2 and make several conjectures concerning the computational complexity of rainbow connection.
A path in an edge-colored graph is said to be a rainbow path if no two edges
on the path have the same color. An edge-colored graph is (strongly) rainbow
connected if there exists a rainbow (geodesic) path between every pair of
vertices. The (strong) rainbow connection number of $G$, denoted by ($scr(G)$,
respectively) $rc(G)$, is the smallest number of colors that are needed in
order to make $G$ (strongly) rainbow connected. Though for a general graph $G$
it is NP-Complete to decide whether $rc(G)=2$, in this paper, we show that the
problem becomes easy when $G$ is a bipartite graph. Moreover, it is known that
deciding whether a given edge-colored (with an unbound number of colors) graph
is rainbow connected is NP-Complete. We will prove that it is still NP-Complete
even when the edge-colored graph is bipartite. We also show that a few NP-hard
problems on rainbow connection are indeed NP-Complete.
Rainbow connection number, rc(G), of a connected graph G is the minimum
number of colors needed to color its edges so that every pair of vertices is
connected by at least one path in which no two edges are colored the same (Note
that the coloring need not be proper). In this paper we study the rainbow
connection number with respect to three important graph product operations
(namely cartesian product, lexicographic product and strong product) and the
operation of taking the power of a graph. In this direction, we show that if G
is a graph obtained by applying any of the operations mentioned above on
non-trivial graphs, then rc(G) <= 2r(G)+c, where r(G) denotes the radius of G
and c \in {0,1,2}. In general the rainbow connection number of a bridgeless
graph can be as high as the square of its radius [Basavaraju et. al, 2010].
This is an attempt to identify some graph classes which have rainbow connection
number very close to the obvious lower bound of diameter (and thus the radius).
The bounds reported are tight upto additive constants. The proofs are
constructive and hence yield polynomial time (2 + 2/r(G))-factor approximation
algorithms.
The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤ r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K
1,n
for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤ rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 − 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1–18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for $rc(L(G))$, in terms of the number of edge-disjoint triangles of $G$ where $L(G)$ is the line graph of $G$. We also give results on the iterated line graphs. Comment: 11 pages
Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex colorings. With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.
Examples of Codes Kraft Inequality Optimal Codes Bounds on the Optimal Code Length Kraft Inequality for Uniquely Decodable Codes Huffman Codes Some Comments on Huffman Codes Optimality of Huffman Codes Shannon–Fano–Elias Coding Competitive Optimality of the Shannon Code Generation of Discrete Distributions from Fair Coins Summary Problems Historical Notes
1. Introduction (Motivation and definitions, Terminology and notations).- 2. (Strong) Rainbow connection number(Basic results, Upper bounds for rainbow connection number, For some graph classes, For dense and sparse graphs, For graph operations, An upper bound for strong rainbow connection number).- 3. Rainbow k-connectivity.- 4. k-rainbow index.- 5. Rainbow vertex-connection number.- 6. Algorithms and computational complexity.- References.
Let G be a nontrivial connected graph on which an edge coloring cE(G)→{1,2,⋯,k}, k∈ℕ, is defined, where adjacent edges may be colored with the same color. A path P in G is a rainbow path if no two edges of P are colored the same. The graph G is rainbow-connected if G contains a rainbow u-v path for every two vertices u and v of G. The minimum k for which there exists such a k-edge coloring is the rainbow connection number rc(G) of G. If for every pair u,v of distinct vertices, G contains a rainbow u-v geodesic, then G is strongly rainbow-connected. The minimum k for which there exists a k-edge coloring of G that results in a strongly rainbow-connected graph is called the strong rainbow connection number src(G) of G. Thus rc(G)≤src(G) for every nontrivial connected graph G. Both rc(G) and src(G) are determined for all complete multipartite graphs G as well as other classes of graphs. For every pair a,b of integers with a≥3 and b≥(5a-6)/3, it is shown that there exists a connected graph G such that rc(G)=a and src(G)=b.
Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex colorings. With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.
We consider the problem of coloring a 3-colorable 3-uniform hypergraph. In the minimization version of this problem, given a 3-colorable 3-uniform hypergraph, one seeks an algorithm to color the hypergraph with as few colors as possible. We show that it is NP-hard to color a 3-colorable 3-uniform hypergraph with constantly many colors. In fact, we show a stronger result that it is NP-hard to distinguish whether a 3-uniform hypergraph with n vertices is 3-colorable or it contains no independent set of size δn for an arbitrarily small constant δ > 0. In the maximization version of the problem, given a 3-uniform hypergraph, the goal is to color the vertices with 3 colors so as to maximize the number of non-monochromatic edges. We show that it is NP-hard to distinguish whether a 3-uniform hypergraph is 3-colorable or any coloring of the vertices with 3 colors has at most 8/9 + ε fraction of the edges nonmonochromatic where ε > 0 is an arbitrarily small constant. This result is tight since assigning a random color independently to every vertex makes 8/9 fraction of the edges non-monochromatic. These results are obtained via a new construction of a probabilistically checkable proof system (PCP) for NP. We develop a new construction of the PCP Outer Verifier. An important feature of this construction is smoothening of the projection maps. Dinur, Regev and Smyth (2002) independently showed that it is NP-hard to color a 2-colorable 3-uniform hypergraph with constantly many colors. In the "good case", the hypergraph they construct is 2-colorable and hence their result is stronger. In the "bad case" however, the hypergraph we construct has a stronger property, namely, it does not even contain an independent set of size δn.
In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems.
We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.
An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edge- connected. We prove that if G has n vertices and minimum degree then rc(G) < 20n= . This solves open problems from (5) and (3). A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if G has n vertices and minimum degree then rvc(G) < 11n= . We note that the proof in this case is dierent from the proof for the edge- colored case, and we cannot deduce one from the other.
The splittance of an arbitrary graph is the minimum number of edges to be added or removed in order to produce a split graph
(i.e. a graph whose vertex set can be partitioned into a clique and an independent set). The splittance is seen to depend
only on the degree sequence of the graph, and an explicit formula for it is derived. This result allows to give a simple characterization
of the degree sequences of split graphs. Worst cases for the splittance are determined for some classes of graphs (the class
of all graphs, of all trees and of all planar graphs).
A path in an edge-colored graph is called a \emph{rainbow path} if all edges
on it have pairwise distinct colors. For $k\geq 1$, the
\emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the
minimum number of colors required to color the edges of $G$ in such a way that
every two distinct vertices are connected by at least $k$ internally disjoint
rainbow paths. In this paper, we study rainbow-$k$-connectivity in the setting
of random graphs. We show that for every fixed integer $d\geq 2$ and every
$k\leq O(\log n)$, $p=\frac{(\log n)^{1/d}}{n^{(d-1)/d}}$ is a sharp threshold
function for the property $rc_k(G(n,p))\leq d$. This substantially generalizes
a result due to Caro et al., stating that $p=\sqrt{\frac{\log n}{n}}$ is a
sharp threshold function for the property $rc_1(G(n,p))\leq 2$. As a
by-product, we obtain a polynomial-time algorithm that makes $G(n,p)$
rainbow-$k$-connected using at most one more than the optimal number of colors
with probability $1-o(1)$, for all $k\leq O(\log n)$ and $p=n^{-\epsilon(1\pm
o(1))}$ for some constant $\epsilon\in[0,1)$.
The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from Schiermeyer (Combinatorial Algorithms, Springer, Berlin/Hiedelberg, 2009, pp. 432–437), improving the previously best known bound of 20n/δ (J Graph Theory 63 (2010), 185–191). This bound is tight up to additive factors by a construction mentioned in Caro et al. (Electr J Combin 15(R57) (2008), 1).
As an intermediate step we obtain an upper bound of 3n/(δ + 1) − 2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree δ. This bound is tight up to an additive constant of 2. This result may be of independent interest.
We also show that for every connected graph G with minimum degree at least 2, the rainbow connection number, rc(G), is upper bounded by Γc(G) + 2, where Γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G) + c, 1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, asteroidal triple-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G)⩽3·radius(G). In most of these cases, we also demonstrate the tightness of the bounds.
Thesis (M.S.) Massachusetts Institute of Technology. Dept. of Electrical Engineering, 1949. Bibliography: leaf 62. by Leon G. Kraft, Jr. M.S.
Thesis (M.S.) Massachusetts Institute of Technology. Dept. of Electrical Engineering, 1954. Includes bibliographical references (leaves 59-60).
Rainbow connectivity of g (n, p) at the connectivity threshold
- A Frieze
- C Tsourakakis
A. Frieze and C.E. Tsourakakis. Rainbow connectivity of g(n, p) at the connectivity
threshold. Arxiv preprint arXiv:1201.4603, 2012.
- M Basavaraju
- L S Chandran
- D Rajendraprasad
- A Ramaswamy
M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy. Rainbow
connection number and radius. Arxiv preprint arXiv:1011.0620v1, 2010.
Rainbow connection number of graph power and graph products
- M Basavaraju
- L S Chandran
- D Rajendraprasad
- A Ramaswamy
M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy. Rainbow connection number of graph power and graph products. Arxiv preprint
arXiv:1104.4190, 2011.