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A red-blue coloring of S 2,5 .
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Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence G1, G2, …, Gk of pairwise edge-disjoint subgraphs of G such that each subgraph Gi (1≤i≤k) is monochromatic and Gi is isomorphic to a proper subgraph of Gi+1 (1≤i≤k−1). The ascending Ramsey index ARc(G) of G with res...
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... double star S 2,5 has a size m = 6 = ( 3+1 2 ) and so k = 3. In the red-blue coloring of S 2,5 shown in Figure 3 where a solid edge indicates a red edge and a thin edge indicates a blue edge, the maximum length of an ascending Ramsey sequence in S 2,5 is 2 and so AR(S 2,5 ) = 2 = k − 1. In fact, AR(S 2,b ) = k − 1 for all double stars S 2,b where b = ( k+1 2 ) − 1 ≥ 5. Proposition 2. If b = ( k+1 2 ) − 1 ≥ 5 for some integer k, then AR(S 2,b ) = k − 1. ...
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A graph $G$ of order $n$ is arbitrarily partitionable~(AP in abbreviation)~if for each sequence $\tau=(n_{1},n_{2},\ldots,n_{k})$ of positive integers such that it sums up to $n$,~there exists a partition $(V_{1},\ldots,V_{k})$ of the vertex set $V$ such that $|V_{i}|=n_{i}$ and $V_{i}$ induces a connected subgraph for all $i=1,\ldots,k$.~A star-li...
Citations
... Suppose that a red-blue coloring of a graph G = mK 2 or G = K 1,m is given, where ( k+1 2 ) ≤ m < ( k+2 2 ). It was shown in [12] that there is not only an ascending subgraph decomposition {G 1 , G 2 , . . ., G k } of G but one in which each subgraph is monochromatic as well. ...
... Consequently, if AR(G) = k for some graph G, then for every red-blue coloring of G, there is a Ramsey chain of length k in G, while there exists at least one red-blue coloring for which there is no Ramsey chain of length greater than k. These concepts were introduced and studied in [11,12], using somewhat different technology, and they were studied further in [24,25]. An immediate observation on Ramsey indexes of graphs was presented in [12]. ...
... These concepts were introduced and studied in [11,12], using somewhat different technology, and they were studied further in [24,25]. An immediate observation on Ramsey indexes of graphs was presented in [12]. Observation 1 ([12]). ...
Every red–blue coloring of the edges of a graph G results in a sequence G1, G2, …, Gℓ of pairwise edge-disjoint monochromatic subgraphs Gi (1≤i≤ℓ) of size i, such that Gi is isomorphic to a subgraph of Gi+1 for 1≤i≤ℓ−1. Such a sequence is called a Ramsey chain in G, and ARc(G) is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index AR(G) of G is the minimum value of ARc(G) among all the red–blue colorings c of G. If G has size m, then k+12≤m<k+22 for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S, AR(G)=k if and only if k+12≤m<k+22. Two of these classes are the matchings mK2 and paths Pm+1 of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with k+12<m<k+22, then AR(F)=k. Furthermore, if F is a linear forest of size k+12, where k≥4, that has at most k−12 components, then AR(F)=k, while for each integer t with k−12<t<k+12 there is a linear forest F of size k+12 with t components, such that AR(F)=k−1.
... Among the observations presented in [3] is the following. On the other hand, if G is a graph of size m such that AR(G) ≥ k, then m ≥ k+1 2 . ...
