Berge–Fulkerson coloring for C(8)‐linked graphs. Issue 1 (8th September 2017)

• Record Type: Journal Article
• Authors: Hao, Rong‐Xia; Zhang, Cun‐Quan; Zheng, Ting
• Is Part Of: Journal of graph theory. Volume 88:Issue 1(2018)
• Page Start: 46
• Page End: 60
• Publication Date: 2017-09-08
• Subjects: Berge–Fulkerson conjecture -- Berge–Fulkerson coloring -- oddness -- perfect matching -- snarks -- 4‐flow; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22184
• Languages: English
• ISSNs: 0364-9024

Abstract

Abstract: It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them . Let G be a cubic graph and F = { C 1, …, C r } be a 2‐factor of G such that | C j | is odd if and only if j ≤ 2 k for some integer k . The 2‐factor F is C (8) ‐linked if, for every i ≤ k, there is a circuit D i of length 8 with edge sequence e 1 i … e 8 i where e 1 i, e 5 i ∈ E ( C 2 i − 1 ) and e 3 i, e 7 i ∈ E ( C 2 i ) . And the cubic graph G is C (8) ‐linked if it contains a C (8) ‐linked 2‐factor. It is proved in this article that every C (8) ‐linked cubic graph is Berge–Fulkerson colorable . It is also noticed that many classical families of snarks (including some high oddness snarks) are C (8) ‐linked. Consequently, the Berge–Fulkerson conjecture is verified for these infinite families of snarks.