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《Discrete Mathematics》2022,345(8):112902
For a simple graph G, denote by n, , and its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if and for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that must be edge-chromatic critical if , and they verified this when . In this paper, we prove it for . 相似文献
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《Discrete Mathematics》2022,345(8):112917
Let and denote the flow number and the circular flow number of a flow-admissible signed graph , respectively. It is known that for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that holds also for every flow-admissible signed graph . This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that if and only if and if , then . We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area. 相似文献
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