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《Discrete Mathematics》2022,345(7):112866
Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then . If G is allowed to be disconnected, then the upper bound for was obtained by Donald [7], which was improved to independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, is reached and so this bound is tight. If triangles are forbidden in G, then can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that , which improves the above result with . 相似文献
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Let GP be the m-Paley graph defined on the finite field with order . We study eigenfunctions and maximal cliques in generalised Paley graphs GP , where . In particular, we explicitly construct maximal cliques of size or in GP , and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue of GP . These new results extend the work of Baker et al. and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP (first proved by Sziklai). 相似文献
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