Abstract
Let G be a graph. The core of G, denoted by G Δ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) denotes the maximum degree of G. A k -edge coloring of G is a function f : E(G) → L such that |L| = k and f (e 1) ≠ f (e 2) for all two adjacent edges e 1 and e 2 of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.
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References
Akbari S., Cariolaro D., Chavoshi M., Ghanbari M., Zare S.: Some Criteria for a graph to be Class 1. Discrete Math. 312, 2593–2598 (2012)
Akbari S., Ghanbari M., Kano M., Nikmehr M.J.: The chromatic index of a graph whose core has maximum degree 2. Electron. J. Comb. 19, P58 (2012)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North Holland, New York (1976)
Cariolaro D., Cariolaro G.: Colouring the petals of a graph. Electron. J. Comb. 10, R6 (2003)
Chetwynd A.G., Hilton A.J.W.: A Δ-subgraph condition for a graph to be Class 1. J. Combin. Theory Ser. B. 46, 37–45 (1989)
Chetwynd A.G., Hilton A.J.W.: 1-factorizing regular graphs of high degree—an improved bound. Discret. Math. 75, 103–112 (1989)
Dugdale J.K., Hilton A.J.W.: A sufficient condition for a graph to be the core of a Class 1 graph. Comb. Probab. Comput. 9, 97–104 (2000)
Fiorini, S., Wilson R.J.: Edge-Colourings of Graphs. Research Notes in Mathematics, Pitman (1977)
Fournier J.-C.: Coloration des arêtes d’un graphe. Cahiers du CERO (Bruxelles) 15, 311–314 (1973)
Hilton A.J.W.: Two conjectures on edge-colouring. Discrete Math. 74, 61–64 (1989)
Hilton A.J.W., Zhao C.: A sufficient condition for a regular graph to be Class 1. J. Graph Theory 17(6), 701–712 (1993)
Hilton A.J.W., Zhao C.: On the edge-colouring of graphs whose core has maximum degree two. J. Comb. Math. Comb. Comput. 21, 97–108 (1996)
Hilton A.J.W., Zhao C.: The chromatic index of a graph whose core has maximum degree two. Discrete Math 101, 135–147 (1992)
Hoffman D.G.: Cores of Class II graphs. J. Graph Theory 20, 397–402 (1995)
Holyer I.: The NP-completeness of edge-colouring. SIAM J. Comput. 10, 718–720 (1981)
Song Z., Yap H.P.: Chromatic index critical graphs of even order with five major vertices. Graph Comb. 21, 239–246 (2005)
Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskret. Analiz. 3, 25–30 (1964) (in Russian)
Vizing, V.G.: Critical graphs with a given chromatic class. Diskret. Analiz. 5, 6–17 (1965) (in Russian)
Zhanga L., Shi W., Huanga X., Li G.: New results on chromatic index critical graphs. Discrete Math. 309, 3733–3737 (2009)
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Akbari, S., Ghanbari, M. & Nikmehr, M.J. The Chromatic Index of a Graph Whose Core is a Cycle of Order at Most 13. Graphs and Combinatorics 30, 801–819 (2014). https://doi.org/10.1007/s00373-013-1317-9
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DOI: https://doi.org/10.1007/s00373-013-1317-9