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Large Rainbow Matchings in Edge-Coloured Graphs
Abstract
A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k ⥠4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least âk/2â. A properly edge-coloured K4 has no such matching, which motivates the restriction k ⥠4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size âk/2â is guaranteed to exist, and they proved several sufficient conditions for a matching of size âk/2â. We prove the conjecture in full.
... It is easy to verify that f K 4 has no a rainbow matching of size 2, which motivates the restrictiondðGÞ ! 4. In particular, the bound of this conjecture is sharp for properly edge-colored complete graphs. This conjecture was partially confirmed in [10] and fully confirmed in [8]. In particular, Kostochka and Yancey [8] proved that if G is not f K 4 , then rðGÞ ! ...
... This conjecture was partially confirmed in [10] and fully confirmed in [8]. In particular, Kostochka and Yancey [8] proved that if G is not f K 4 , then rðGÞ ! ddðGÞ=2e. ...
... Let n :¼ jV ðGÞj. By the result of Kostochka and Yancey [8] and Theorem 3, we may assume that n ! 2k þ 1 and dðGÞ k À 1. ...
A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of distinct colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(3), 2020] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|≥12k2+4k, or G is a properly edge-colored graph with |V(G)|≥8k. In this paper, we show that every edge-colored graph G with |V(G)|≥4k-4 and average color degree at least 2k-1 contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least 2k-1 contains a rainbow matching of size at least k. The bound is sharp for complete graphs.
... It is easy to verify that f K 4 has no a rainbow matching of size 2, which motivates the restrictiondðGÞ ! 4. In particular, the bound of this conjecture is sharp for properly edge-colored complete graphs. This conjecture was partially confirmed in [10] and fully confirmed in [8]. In particular, Kostochka and Yancey [8] proved that if G is not f K 4 , then rðGÞ ! ...
... This conjecture was partially confirmed in [10] and fully confirmed in [8]. In particular, Kostochka and Yancey [8] proved that if G is not f K 4 , then rðGÞ ! ddðGÞ=2e. ...
... Let n :¼ jV ðGÞj. By the result of Kostochka and Yancey [8] and Theorem 3, we may assume that n ! 2k þ 1 and dðGÞ k À 1. ...
A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|>=12k^2+4k, or G is a properly edge-colored graph with |V(G)|>=8k. In this paper, we show that every edge-colored graph G with |V(G)|>=4k-4 and average color degree at least 2k-1 contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least 2k-1 contains a rainbow matching of size at least k. The bound is sharp for complete graphs.
... For s = 1, the answer is negative: let G be a complete graph K m+1 (m is even) which is an edge disjoint union of m 1-factors, however, the size of maximum matching is at most m 2 . Indeed, it is best possible, see [11]. But how about when we restrict ourselves to large graph? ...
... Recently, the study of rainbow paths and cycles under minimum color degree condition has received much attention, see [6,15]. For rainbow matchings under minimum color degree condition, see [11,10,16,13,14,19]. ...
... By (2.2), it follows that 5n − 16p ≤ 4q. By assumption n ≥ 3.2m − 1 = 3.2p + 2.2, so we finally arrive at 11 4 ≤ q. Since q is an integer, we have q ≥ 3. ...
... We show that for a strongly edge-colored graph G, if |V (G)| solvable, deciding whether an edge-colored graph has a maximum rainbow matching of size at least k is an NP-Complete problem, mentioned in Garey and Johnson [2] as the Multiple Choice Matching problem. There have been several studies giving lower bounds for the size of maximum rainbow matchings in edge-colored graphs [11,6,5,7]. Rainbow matchings in properly edge-colored graphs were studied in connection with the famous Latin square transversal problem. ...
... Recently, the results in this area have focused on lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs of size smaller than 4δ(G) − 4. From a result of Kostochka and Yancey [5] for arbitrary edge-colored graphs, it follows that if G is a properly edge-colored graph that is not K 4 , then G contains a rainbow matching of size ...
... Wang [10] showed that if G is a properly edge-colored graph with |V (G)| ≥ 8δ(G) 5 , then G has a rainbow matching of size at least 3δ(G) 5 ...
A rainbow matching of an edge-colored graph is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph in terms of its minimum degree . Wang (2011) asked whether there exists a function such that a properly edge-colored graph with at least vertices is guaranteed to contain a rainbow matching of size . This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is , for and , for . Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than vertices. Strong edge-coloring of a graph is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph in terms of . We show that for a strongly edge-colored graph , if , then has a rainbow matching of size , and if , then has a rainbow matching of size . In addition, we prove that if is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least .
... Kostochka, Pfender and Yancey [5] showed that every (not necessarily properly) edge-coloured G on n ≥ 17k 2 /4 vertices with δ c (G) ≥ k contains a rainbow matching of size k. Tan and the author [9] improved the bound to n ≥ 4k − 4 for k ≥ 4. In this paper we show that n ≥ 7k/2 + 2 is sufficient. ...
... By Vizing's theorem, |S| ≤ 2(i − 1)(∆(G i ) + 1) ≤ 2(i − 1)(t + 1). By (5) and the assumption on i, we have |N H (S)| ≥ (t − 4)n/2 ≥ 2(i − 1)(t + 1) ≥ |S|. Therefore, Hall's condition holds for this case. ...
Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph
such that its edges have distinct colours. The minimum colour degree
$\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges
incident with a vertex of $G$. We show that every edge-coloured graph $G$ on
$n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching
of size at least $k$, which improves the previous result for $k \ge 10$.
Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same
colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and
$\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most
$\lfloor tn/2 \rfloor $ rainbow matchings. This result is sharp and improves a
result of LeSaulnier and West.
... The color degree of a vertex v in an edge-colored graph G, writtenˆdwrittenˆ writtenˆd(v), is the number of different colors on edges incident to v. We letˆδletˆ letˆδ(G) denote the minimum color degree among the vertices in G. Wang and Li [7] proved that every edge-colored graph G contains a rainbow matching of size at least 5 ˆ δ(G)−3 12 , and conjectured thatˆδ thatˆ thatˆδ(G)/2 could be guaranteed whenˆδwhenˆ whenˆδ(G) ≥ 4. LeSaulnier et al. [4] then proved that every edge-colored graph G contains a rainbow matching of sizê sizê δ(G)/2 . Finally, Kostochka and Yancey [3] proved the conjecture of Wang and Li in full, and also that triangle-free graphs have rainbow matchings of size ...
... , then G contains a rainbow matching of size δ(G). The proofs of Theorems 1 and 2 depend on the implementation of a greedy algorithm, a significantly different approach than those found in [3], [4], [6], and [7]. This algorithm generates a rainbow matching in a properly edge-colored graph G in O(δ(G)|V (G)| 2 )-time. ...
A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(δ) such that a properly edge-colored graph G with minimum degree δ and order at least f(δ) must have a rainbow matching of size δ. We answer this question in the affirmative; an extremal approach yields that f(δ)=98δ/23<4·27δ suffices. Furthermore, we give an O(δ(G)|V(G)| 2 )-time algorithm that generates such a matching in a properly edge-colored graph of order at least 6·5δ.
... They further conjectured and asked whether one could prove the following Ravsky [20] proposed a claim connecting this problem to rainbow matchings. Using a result of Kostochka and Yancey [11], he showed that the dimension of an n vertex experiment graph is at most n − 2. Neugebauer [17] used these ideas and did computational experiments. ...
Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory and have several applications in quantum communication and cryptography. Motivated by this, physicists have been designing various experiments to create high-dimensional GHZ states using multiple entangled particles. In 2017, Krenn, Gu and Zeilinger discovered a bridge between experimental quantum optics and graph theory. A large class of experiments to create a new GHZ state are associated with an edge-coloured edge-weighted graph having certain properties. Using this framework, Cervera-Lierta, Krenn, and Aspuru-Guzik proved using SAT solvers that through these experiments, the maximum dimension achieved is less than $3,4$ using $6,8$ particles, respectively. They further conjectured that using $n$ particles, the maximum dimension achievable is less than $\dfrac{n}{{2}}$ [Quantum 2022]. We make progress towards proving their conjecture by showing that the maximum dimension achieved is less than $\dfrac{n}{\sqrt{2}}$.
... . Kostochka and Yancey [9] completed a proof of Wang and Li's conjecture. Letting n be the size of vertices of a graph G, Lo [10] showed that an edge-colored graph G contains a rainbow matching of size at least k, where k = min δ (G), 2n−4 7 . ...
Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n.
... Just requiring that every available color is used at least once, Fujita, Kaneko, Schiermeyer, and Suzuki [7] studied the existence of large rainbow matchings in edge-colored complete graphs. Kostochka and Yancey [11] showed the existence of large rainbow matchings for edge-colored graphs provided that every vertex is incident with edges of many different colors. ...
For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every $k$-edge colored color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$. For a perfect matching $M$ of $K_{2kn}$, a natural measure for the total deviation of $M$ from being color-balanced is $f(M)=\sum\limits_{i=1}^k|m_i(M)-n|$. While not every color-balanced complete graph $K_{2kn}$ has a color-balanced perfect matching $M$, that is, a perfect matching with $f(M)=0$, we prove the existence of a perfect matching $M$ with $f(M)=O\left(k\sqrt{kn\ln(k)}\right)$ for general $k$ and $f(M)\leq 2$ for $k=3$; the case $k=2$ has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.
... . Kostochka and Yancey [9] completed a proof of Wang and Li's conjecture. Letting n be the size of vertices of a graph G, Lo [10] showed that an edge-colored graph G contains a rainbow matching of size at least k, where k = min δ (G), 2n−4 7 . ...
Given an edge-colored complete graph $K_n$ on $n$ vertices, a perfect (respectively, near-perfect) matching $M$ in $K_n$ with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of $K_n$ by circular distance, and we denote the resulting complete graph by $K^{\bullet}_n$. We show that when $K^{\bullet}_n$ has an even number of vertices, it contains a rainbow perfect matching if and only if $n=8k$ or $n=8k+2$, where $k$ is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in $K^{\bullet}_n$. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in $K^{\bullet}_n$.
... LeSaulnier and Stocker et al. [16] gave sufficient conditions for a rainbow matching of size at least k/2 that holds for even value of k. Kostochka and Yancey [14] proved the conjecture of Wang and Li in full. Lo [17] showed that every edge-coloured graph on n ≥ 7k/2 + 2 vertices with color degree at least k contains a rainbow matching of size at least k. ...
In this paper, we study the NP-complete colorful variant of the classic matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, the goal is to decide whether there exists a matching of size at least k such that the edges in the matching have distinct colors. Previously, in [MFCS’17], we studied this problem from the view point of Parameterized Complexity and gave efficient FPT algorithms as well as a quadratic kernel on paths. In this paper we design a quadratic vertex kernel for Rainbow Matching on general graphs; generalizing the earlier quadratic kernel on paths to general graphs. For our kernelization algorithm we combine a graph decomposition method with an application of expansion lemma.
... In [5], Li and Wang conjectured that any graph withˆδwithˆ withˆδ(G) ≥ m ≥ 4 contains a rainbow matching of size ⌈ m 2 ⌉. This conjecture was partially confirmed in [4], and fully confirmed in [3]. ...
The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper, generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$; These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.
... 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.
... One often asks for the size of the largest rainbow matching in an edge-colored graph with certain restrictions (see, e.g., [4,14,15,22]). In this paper, we consider the complexity of this problem. ...
A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem.
We show, among others, that
* can be approximated by a polynomial algorithm with approximation ratio
$2/3-\eps$. * is APX-complete, even when restricted to properly edge-colored
linear forests without a $5$-vertex path, and is solvable in %time $O(m^{3/2})$
on edge-colored $m$-edge polynomial time for edge-colored forests without a
$4$-vertex path. * is APX-complete, even when restricted to properly
edge-colored trees without an $8$-vertex path, and is solvable in %time
$O(n^{7/2})$ on edge-colored $n$-vertex polynomial time for edge-colored trees
without a $7$-vertex path. * is APX-complete, even when restricted to properly
edge-colored paths.
These results provide a dichotomy theorem for the complexity of the problem
on forests and trees in terms of forbidding paths. The latter is somewhat
surprising, since, to the best of our knowledge, no (unweighted) graph problem
prior to our result is known to be NP-hard for simple paths. We also address
the parameterized complexity of the problem.
... During the last decade the problem of finding conditions for large rainbow matchings in a graph was extensively explored. See, for example, [1,8,10,11]. Many results and conjectures on the subject were influenced by the well-known conjectures of Ryser [12], asserting that every Latin square of odd order n has a transversal of order n, and Brualdi [6] (see also [5] p. 255), asserting that every Latin square of even order n has a partial transversal of size n − 1. Brualdi's conjecture may also be casted into the form of a rainbow matching problem: Conjecture 1.1. ...
Let g(n) be the least number such that every collection of n matchings, each of size at least g(n), in a bipartite graph, has a full rainbow matching. R. Aharoni and E. Berger [Electron. J. Comb. 16, No. 1, Research Paper R119, 9 p. (2009; Zbl 1186.05118)] conjectured that g(n)=n+1 for every n>1. This generalizes famous conjectures of Ryser, Brualdi and Stein. Recently, R. Aharoni et al. [“On a generalization of the Ryser-Brualdi-Stein conjecture”, manuscript] proved that g(n)≤⌊7 4n⌋. We prove that g(n)≤⌊5 3n⌋.
... A problem of rainbow Hamilton cycles in colored complete graphs was mentioned by Erdös, Nešetřil and Rödl [6], and later studied by Hahn and Thomassen [8], Frieze and Reed [7] and Albert, Frieze and Reed [1], respectively. Rainbow matchings were studied by Wang and H. Li [16], Lesaulnier et al. [11], and Kostochka and Yancey [10]. Chen and X. Li [4,5] studied the existence of long rainbow paths. ...
In this paper we obtain a new sufficient condition for the existence of
directed cycles of length 4 in oriented bipartite graphs. As a corollary, a
conjecture of H. Li is confirmed. As an application, a sufficient condition for
the existence of rainbow cycles of length 4 in bipartite edge-colored graphs is
obtained.
... In [3,2] conditions were considered in terms of lower bounds on the size of the largest matching in an auxiliary graph. For additional problems and results on rainbow matchings, the interested reader is referred to [14,3,2,13,5]. ...
Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is
a collection of t disjoint edges, each having a different color. In this note
we study the problem of finding a rainbow $t$-matching in an r-partite
r-uniform multi-hypergraph whose edges are colored with f colors such that
every color class is a matching of size t. This problem was posed by Aharoni
and Berger, who asked to determine the minimum number of colors which
guarantees a rainbow matching. We improve on the known upper bounds for this
problem for all values of the parameters. In particular for every fixed r, we
give an upper bound which is polynomial in t, improving the superexponential
estimate of Alon. Our proof also works in the setting not requiring the
hypergraph to be r-partite.
... Kostochka, Pfender and Yancey [6] considered a similar problem with δ c (G) instead of properly edge-coloured graphs. They showed that if G is such that δ c (G) ≥ k and n > 17 4 k 2 , then G contains a rainbow matching of size k. ...
A rainbow subgraph in an edge-coloured graph is a subgraph such that its
edges have distinct colours. The minimum colour degree of a graph is the
smallest number of distinct colours on the edges incident with a vertex over
all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured
graph on $n$ vertices with minimum colour degree at least $k$ contains a
rainbow matching of size at least $k$, provided $n\geq (17/4)k^2$. In this
paper, we show that $n\geq 4k-4$ is sufficient for $k \ge 4$.
... LeSaulnier et al. [13] proved that r(G, φ) ≥ k 2 for general graphs, and gave several conditions sufficient for a rainbow matching of size k 2 . In [11], the conjecture was proved in full. The only known extremal examples for the bound have at most k + 2 vertices. ...
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose
edges have distinct colors. The \textit{color degree} of a vertex $v$ is the
number of different colors on edges incident to $v$. We show that if $n$ is
large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with
minimum color degree at least $k$ contains a rainbow matching of size at least
$k$.
... 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 a not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges of G of color i, and call G color-balanced if mi(G)=mj(G) for every two colors i and j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly n-edge colored complete bipartite graph Kn,n has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M. For a perfect matching M of K2kn, a natural measure for the total deviation of M from being color-balanced is f(M)=∑i=1k|mi(M)−n|. While not every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M, that is, a perfect matching with f(M)=0, we prove the existence of a perfect matching M with f(M)=O(kknln(k)) for general k and f(M)≤2 for k=3; the case k=2 has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.
For efficient design of parallel algorithms on multiprocessor architectures with memory banks, simultaneous access to a specified subgraph of a graph data structure by multiple processors requires that the data items belonging to the subgraph reside in distinct memory banks. Such “conflict-free” access to parallel memory systems and other applied problems motivate the study of rainbow coloring of a graph, in which there is a fixed template T (or a family of templates), and one seeks to color the vertices of an input graph G with as few colors as possible, so that each copy of T in G is rainbow colored, i.e., has no two vertices the same color. In the above example, the data structure is modeled as the host graph G, and the specified subgraph as the template T. We call such coloring a template-driven rainbow coloring (or TR-coloring). For large data sets, it is also important to ensure that no memory bank (color) is overloaded, i.e., the coloring is as balanced as possible. Additionally, for fast access to data, it is desirable to quickly determine the address of a memory bank storing a data item. For arbitrary topology of G and T, finding an optimal and balanced TR-coloring is a challenging problem. This paper focuses on rainbow coloring of proper interval graphs (as hosts) for cycle templates. In particular, we present an O(k·|V|+|E|) time algorithm to find a TR-coloring of a proper interval graph G with respect to k-length cycle template, Ck. Our algorithm produces a coloring that is (i) optimal, i.e., it uses minimum possible number of colors in any TR-coloring; (ii) balanced, i.e, the vertices are evenly distributed among the different color classes; and (iii) explicit, i.e., the color assigned to a vertex can be computed by a closed form formula in constant time.
Let Gϕ be an edge-colored graph with order n and δGϕ be the smallest number of distinctly colored edges incident to a vertex of Gϕ. To determine the smallest function h(n) such that any edge-colored graph Gϕ of order n contains a rainbow cycle if δGϕ≥h(n)? It is known that log2n<h(n)≤n+12. For a positive integer r, an edge-coloring ϕ of Gϕ is called r-good if each color appears at most r times at each vertex to generalize the proper edge-coloring of Gϕ. In this paper, for any given ϵ ∈ (0, 1), we show that there exists a constant C(ϵ,r) such that h(n)<C(ϵ,r)nϵ for any r-good edge-colored graph Gϕ.
The Ryser Conjecture which states that there is a transversal of size n in a Latin square of odd order n is equivalent to finding a rainbow matching of size n in a properly edge-colored Kn,n using n colors when n is odd. Let δ be the minimum degree of a graph. Wang proposed a more general question to find a function f(δ) such that every properly edge-colored graph of order f(δ) contains a rainbow matching of size δ which currently has the best bound of f(δ)≤3.5δ+2 by Lo. Babu, Chandran and Vaidyanathan investigated Wang's question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least 2δ+2 has a rainbow matching of size δ. In this note, we extend this result to graphs of order at least 2δ+1.
The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph G is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. in this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.
A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdős and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.
The famous Ryser Conjecture states that there is a transversal of size in a latin square of odd order , which is equivalent to finding a rainbow matching of size in a properly edge-colored when is odd. Let denote the minimum degree of a graph. In 2011, Wang proposed a more general question to find a function ( ) such that for each properly edge-colored graph of order , there exists a rainbow matching of size . The best bound so far is due to Lo. Babu et al. considered this problem in strongly edge-colored graphs in which each path of length 3 is rainbow. They proved that if is a strongly edge-colored graph of order at least , then has a rainbow matching of size . They proposed an interesting question: Is there a constant greater than such that every strongly edge-colored graph has a rainbow matching of size at least if ? Clearly, . We prove that if is a strongly edge-colored graph with minimum degree and order at least , then has a rainbow matching of size .
Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G. This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G–the minimum number of distinct colors occurring at edges incident to any vertex of G–denoted by ϑ(G).
Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be ⌈2ϑ(G)3⌉. Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least ϑ(G)−1, if 1≤ϑ(G)≤7, and at least ⌈3ϑ(G)5⌉+1, if ϑ(G)≥8. They conjectured that the tight lower bound would be ϑ(G)−1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if ϑ(G)≥8, then G contains a heterochromatic path of length at least ⌈2ϑ(G)3⌉+1.
In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least ϑ(G)−o(ϑ(G)) and if the girth of G is at least 4log2(ϑ(G))+2, then it contains a heterochromatic path of length at least ϑ(G)−2, which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of ⌊5ϑ(G)6⌋ and for bipartite graphs we obtain a lower bound of ⌈6ϑ(G)−37⌉.
In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least ⌊13ϑ(G)17⌋. This improves the previously known ⌈3ϑ(G)4⌉ bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least ⌈2ϑ(G)3⌉.
Let G be an edge-colored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edge-chromatic number of G, written ˆχ ′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is t-tolerant if it contains no monochromatic star with t+1 edges. If G is t-tolerant, then ˆχ ′(G) < t(t + 1)n ln n, and examples exist with ˆχ ′(G) ≥ t 2 (n − 1). The rainbow domination number, written ˆγ(G), is the minimum number of disjoint rainbow stars needed to cover V (G). For t-tolerant edge-colored n-vertex graphs, we generalize 1+ln k δ(G) classical bounds on the domination number: (1) ˆγ(G) ≤ k n (where k
Given sets $F_1, \ldots,F_n$, a {\em partial rainbow function} is a partial
choice function of the sets $F_i$. A {\em partial rainbow set} is the range of
a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if
$M$ and $N$ are matroids on the same ground set, and $F_1, \ldots,F_n$ are
pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a
rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of
Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a
rainbow set of size at least $n-\sqrt{n}$.
A graph is properly edge-colored if no two adjacent edges have the same
color. The smallest number of edges in a graph any of whose proper edge
colorings contains a totally multicolored copy of a graph H is the size
anti-Ramsey number AR_s(H) of H. This number in offline and online setting is
investigated here.
Let G be a properly edge-colored graph. A rainbow matching of G is a matching in which no two edges have the same color. Let δ denote the minimum degree of G. We show that if |V(G)| > (δ
2 + 14δ + 1)/4, then G has a rainbow matching of size δ, which answers a question asked by G. Wang [Electron. J. Combin., 2011, 18: #N162] affirmatively. In addition, we prove that if G is a properly colored bipartite graph with bipartition (X, Y) and max{|X|, |Y|} > (δ
2 + 4δ − 4)/4, then G has a rainbow matching of size δ.
Nowadays the term monochromatic and heterochromatic (or rain-bow, multicolored) subgraphs of an edge colored graph appeared fre-quently in literature, and many results on this topic have been ob-tained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochro-matic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and het-erochromatic subgraphs. We have to point out that there are a lot of results of Ramsey type problem on monochromatic and heterochro-matic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not contain them, either.
A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex v is the number of different colors on edges incident to v. Wang and Li conjectured that for k ≥ 4, every edge-colored graph with minimum color degree at least k contains a rainbow matching of size at least ⌈k/2⌉. We prove the slightly weaker statement that a rainbow matching of size at least ⌊k/2⌋ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least ⌈k/2⌉ that fail to hold only for finitely many exceptions (for each odd k).
El-Zanati et al proved that for any 1-factorization $\mathcal{F}$ of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor.
Let (G,C) be an (edge-)colored bipartite graph. A heterochromatic matching of G is such a matching in which no two edges have the same color. Let d c (v), named the color degree of a vertex v, be defined as the maximum number of edges adjacent to v, that have distinct colors. We show that if d c (v)≥k≥3 for every vertex v of G, then G has a heterochromatic matching with cardinality at least 2k 3.
Let G be an (edge-)colored graph. A heterochromatic matching of G is a matching in which no two edges have the same color. For a vertex v, let d(c)(v) be the color degree of v. We show that if d(c)(v) >= k for every vertex v of G, then G has a heterochromatic matching of size inverted right perpendicular 5k-3/12 inverted left perpendicular. For a colored bipartite graph with bipartition (X, Y), we prove that if it satisfies a Hall-like condition, then it has a heterochromatic matching of cardinality inverted right perpendicular vertical bar X vertical bar/2 inverted left perpendicular, and we show that this bound is best possible.
Rainbow matching in edge-colored graphs
- T D Lesaulnier
- C Stocker
- P S Wegner
- D B West
T. D. LeSaulnier, C. Stocker, P. S. Wegner, and D. B. West, Rainbow Matching in Edge-Colored Graphs. Electron. J. Combin. 17 (2010), Paper #N26.