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《Discrete Mathematics》2022,345(8):112903
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number is the least integer k for which G admits a coloring with k colors such that each color class induces a -degenerate subgraph of G. So is the chromatic number and is the point arboricity. The point partition number with was introduced by Lick and White. A graph G is called -critical if every proper subgraph H of G satisfies . In this paper we prove that if G is a -critical graph whose order satisfies , then G can be obtained from two non-empty disjoint subgraphs and by adding t edges between any pair of vertices with and . Based on this result we establish the minimum number of edges possible in a -critical graph G of order n and with , provided that and t is even. For the corresponding two results were obtained in 1963 by Tibor Gallai. 相似文献
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《Discrete Mathematics》2022,345(10):113004
Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, can be partitioned into A and B such that is perfect and . We use and to denote a path and a cycle on t vertices, respectively. For two disjoint graphs and , we use to denote the graph with vertex set and edge set , and use to denote the graph with vertex set and edge set . In this paper, we prove that (i) -free graphs are perfectly divisible, (ii) if G is -free with , (iii) if G is -free, and (iv) if G is -free. 相似文献
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《Discrete Mathematics》2022,345(3):112717
A transversal set of a graph G is a set of vertices incident to all edges of G. The transversal number of G, denoted by , is the minimum cardinality of a transversal set of G. A simple graph G with no isolated vertex is called τ-critical if for every edge . For any τ-critical graph G with , it has been shown that by Erd?s and Gallai and that by Erd?s, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to . In this paper, we prove stronger results via spectrum. Let G be a τ-critical graph with and , and let denote the largest eigenvalue of the adjacency matrix of G. We show that with equality if and only if G is , , or , where ; and in particular, with equality if and only if G is . We then apply it to show that for any nonnegative integer r, we have and characterize all extremal graphs. This implies a pure combinatorial result that , which is stronger than Erd?s-Hajnal-Moon Theorem and Gyárfás-Lehel Theorem. We also have some other generalizations. 相似文献
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《Discrete Mathematics》2021,344(12):112604
A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index of G is Δ or . A graph G is class 1 if , and class 2 if ; G is Δ-critical if it is connected, class 2 and for every . A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, for every . Such graphs have intimate relation to -co-critical graphs, where a non-complete graph G is -co-critical if there exists a k-coloring of such that G does not contain a monochromatic copy of but every k-coloring of contains a monochromatic copy of for every . We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all -co-critical graphs. We prove that if G is a -co-critical graph on vertices, then where ε is the remainder of when divided by 2. This bound is best possible for all and . 相似文献
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《Discrete Mathematics》2022,345(11):113058
Given an undirected graph , a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of G, denoted by .In previous work [WG 2020], we showed the upper bound , where denotes the distance to cluster parameter of G. In this paper, we obtain the improved upper bound of . We also exhibit a family of graphs for which , thereby demonstrating that our upper bound is tight. 相似文献
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