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

Akbari, S ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.4153/CMB-2012-046-3
  3. Publisher: 2013
  4. Abstract:
  5. Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by x0(G). It is well known that δ(G) ≤ x0(G) ≤ δ(G) + 1, for any graph G, whereδ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x0(G) = δ(G) and class 2 if x0(G) = δ(G)+1. Also, Gδ is the induced subgraph on all vertices of degreeδ(G). Let f : V(G) ! N be a function. An f -coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v 2 V(G) at most f (v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by x0f (G). It was shown that for every graph G,δf (G) ≤ x0f (G) ≤ δf (G)+1, whereδf (G) = maxv2V(G)ddG(v)= f (v)e. A graph G is said to be f -class 1 if x0f (G) = δf (G), and f -class 2, otherwise. Also, Gδf is the induced subgraph of G on fv 2 V(G) : dG(v)= f (v) = δf (G)g. Hilton and Zhao showed that if Gδ has maximum degree two and G is class 2, then G is critical, Gδ is a disjoint union of cycles and (G) = δ(G)1, where (G) denotes the minimum degree of G, respectively. In this paper, we generalize this theoremto f -coloring of graphs. Also, we determine the f -chromatic index of a connected graph G with jGδf j ≤ 4
  6. Keywords:
  7. F -class 1 ; F -coloring ; F -core
  8. Source: Canadian Mathematical Bulletin ; Volume 56, Issue 3 , 2013 , Pages 449-458 ; 00084395 (ISSN)
  9. URL: http://cms.math.ca/10.4153/CMB-2012-046-3