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A coloring (partition) of the collection of all -subsets of a set is -regular if the number of times each element of appears in each color class (all sets of the same color) is the same number . We are interested in finding the conditions under which a given -regular coloring of is extendible to an -regular coloring of for and . The case was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for . The case was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases and were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases and . 相似文献
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Daniel W. Cranston 《Journal of Graph Theory》2019,92(4):460-487
A graph is -choosable if given any list assignment with for each there exists a function such that and for all , and whenever vertices and are adjacent . Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture. 相似文献
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The distinguishing index of a graph is the least cardinal number such that has an edge-coloring with colors, which is preserved only by the trivial automorphism. We prove a general upper bound for any connected infinite graph with finite maximum degree . This is in contrast with finite graphs since for every there exist infinitely many connected, finite graphs with . We also give examples showing that this bound is sharp for any maximum degree . 相似文献