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Some results on the f-chromatic index of graphs whose f-core has maximum degree 2

Akbari, S ; Sharif University of Technology | 2019

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  1. Type of Document: Article
  2. Publisher: University of Queensland , 2019
  3. Abstract:
  4. Let G be a graph and f: V (G) → ℕ be a function. An f-coloring of a graph G is an edge coloring such that each color appears at each vertex v ∈ 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 χʹf(G). It was shown that for every graph G, Δf(G) ≤ χʹf(G) ≤ Δf(G) + 1, where Δf(G) = maxv ∈V (G)(formula presented). A graph G is said to be f-Class 1 if χʹf(G) = Δf(G), and f-Class 2, otherwise. Also, GΔfis the induced subgraph of G on (formula presented). In this paper, we show that if G is a connected graph with Δ(GΔf) ≤ 2 and G has an edge cut of size at most Δf(G) − 2 which is a star, then G is f-Class 1. Also, we prove that if G is a connected graph and every connected component of GΔfis a unicyclic graph or a tree and GΔfis not 2-regular, then G is f-Class 1. Moreover, we show that except one graph, every connected claw-free graph G whose f-core is 2-regular with a vertex v such that f(v) ≠ 1 is f-Class 1. © The author(s)
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  7. Source: Australasian Journal of Combinatorics ; Volume 75 , 2019 , Pages 32-49 ; 10344942 (ISSN)
  8. URL: https://ajc.maths.uq.edu.au/pdf/75/ajc_v75_p032.pdf