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The chromatic index of a claw-free graph whose core has maximum degree 2

Akbari, S ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1007/s00373-014-1417-1
  3. Publisher: Springer-Verlag Tokyo , 2015
  4. Abstract:
  5. Let $$G$$G be a graph. The core of $$G$$G, denoted by $$G_{Delta }$$GΔ, is the subgraph of $$G$$G induced by the vertices of degree $$Delta (G)$$Δ(G), where $$Delta (G)$$Δ(G) denotes the maximum degree of $$G$$G. A $$k$$k-edge coloring of $$G$$G is a function $$f:E(G)ightarrow L$$f:E(G)→L such that $$|L| = k$$|L|=k and $$f(e_1)e f(e_2)$$f(e1)≠f(e2), for any two adjacent edges $$e_1$$e1 and $$e_2$$e2 of $$G$$G. The chromatic index of $$G$$G, denoted by $$chi '(G)$$χ′(G), is the minimum number $$k$$k for which $$G$$G has a $$k$$k-edge coloring. A graph $$G$$G is said to be Class $$1$$1 if $$chi '(G) = Delta (G)$$χ′(G)=Δ(G) and Class $$2$$2 if $$chi '(G) = Delta (G) + 1$$χ′(G)=Δ(G)+1. Hilton and Zhao conjectured that if $$G$$G is a connected graph, $$Delta (G_{Delta })le 2$$Δ(GΔ)≤2, and $$G$$G is not the Petersen graph with one vertex removed, then $$G$$G is Class $$2$$2 if and only if $$G$$G is overfull. In this paper, we prove this conjecture for claw-free graphs of even order
  6. Keywords:
  7. Class 1 ; Claw-free ; Core ; Edge coloring
  8. Source: Graphs and Combinatorics ; Volume 31, Issue 4 , July , 2015 , Pages 805-811 ; 09110119 (ISSN)
  9. URL: http://link.springer.com/article/10.1007%2Fs00373-014-1417-1