Abstract
Let \(G\) be a graph. The core of \(G\), denoted by \(G_{\Delta }\), is the subgraph of \(G\) induced by the vertices of degree \(\Delta (G)\), where \(\Delta (G)\) denotes the maximum degree of \(G\). A \(k\)-edge coloring of \(G\) is a function \(f:E(G)\rightarrow L\) such that \(|L| = k\) and \(f(e_1)\ne f(e_2)\), for any two adjacent edges \(e_1\) and \(e_2\) of \(G\). The chromatic index of \(G\), denoted by \(\chi '(G)\), is the minimum number \(k\) for which \(G\) has a \(k\)-edge coloring. A graph \(G\) is said to be Class \(1\) if \(\chi '(G) = \Delta (G)\) and Class \(2\) if \(\chi '(G) = \Delta (G) + 1\). Hilton and Zhao conjectured that if \(G\) is a connected graph, \(\Delta (G_{\Delta })\le 2\), and \(G\) is not the Petersen graph with one vertex removed, then \(G\) is Class \(2\) if and only if \(G\) is overfull. In this paper, we prove this conjecture for claw-free graphs of even order.
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Acknowledgments
The authors are grateful to the referee for useful comments. The first and the second authors are indebted to the School of Mathematics, Institute for Research in Fundamental Sciences (IPM) for support. The research of the first and the second authors were in part supported by a grant from IPM (No. 92050212) and (No. 92050014), respectively.
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Akbari, S., Ghanbari, M. & Ozeki, K. The Chromatic Index of a Claw-Free Graph Whose Core has Maximum Degree \(2\) . Graphs and Combinatorics 31, 805–811 (2015). https://doi.org/10.1007/s00373-014-1417-1
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DOI: https://doi.org/10.1007/s00373-014-1417-1