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21.
Let be a cycle of order 4. Write for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has vertices that have no subgraph isomorphic to . In this paper, we show that , where and is a prime number. Note that from Tait and Timmons's works. Since for every integer , one can find a prime such that , we obtain that . 相似文献
22.
We show that there is no (95, 40, 12, 20) strongly regular graph and, consequently, there is no (96, 45, 24, 18) strongly regular graph, no nontrivial regular two‐graph on 96 vertices, and no partial geometry pg(4, 9, 2). The main idea of the result is based on the star complement technique and requires a moderate amount of computation. 相似文献
23.
24.
Bhalchandra D. Thatte 《Journal of Graph Theory》2020,93(2):181-202
Previously we showed that many invariants of a graph can be computed from its abstract induced subgraph poset, which is the isomorphism class of the induced subgraph poset, suitably weighted by subgraph counting numbers. In this paper, we study the abstract bond lattice of a graph, which is the isomorphism class of the lattice of distinct unlabelled connected partitions of a graph, suitably weighted by subgraph counting numbers. We show that these two abstract posets can be constructed from each other except in a few trivial cases. The constructions rely on certain generalisations of a lemma of Kocay in graph reconstruction theory to abstract induced subgraph posets. As a corollary, trees are reconstructible from their abstract bond lattice. We show that the chromatic symmetric function and the symmetric Tutte polynomial of a graph can be computed from its abstract induced subgraph poset. Stanley has asked if every tree is determined up to isomorphism by its chromatic symmetric function. We prove a counting lemma, and indicate future directions for a study of Stanley's question. 相似文献
25.
26.
《Discrete Mathematics》2019,342(4):1186-1190
27.
Andrey A. Dobrynin 《Discrete Mathematics》2019,342(1):74-77
Distance between two vertices is the number of edges in a shortest path connecting them in a connected graph . The transmission of a vertex is the sum of distances from to all the other vertices of . If transmissions of all vertices are mutually distinct, then is a transmission irregular graph. It is known that almost no graphs are transmission irregular. Infinite families of transmission irregular trees of odd order were presented in Alizadeh and Klav?ar (2018). The following problem was posed in Alizadeh and Klav?ar (2018): do there exist infinite families of transmission irregular trees of even order? In this article, such a family is constructed. 相似文献
28.
《Discrete Mathematics》2019,342(2):339-343
A strong edge-coloring of a graph is a partition of its edge set into induced matchings. Let be a connected planar graph with girth and maximum degree . We show that either is isomorphic to a subgraph of a very special -regular graph with girth , or has a strong edge-coloring using at most colors. 相似文献
29.
《Discrete Mathematics》2019,342(10):2834-2842
30.
《Discrete Mathematics》2019,342(7):1956-1965