Berge–Fulkerson coloring for C₍₈₎‐linked graphs

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={C1,…,Cr} be a 2‐factor of G such that false|Cjfalse| is odd if and only if j≤2k for some integer k. The 2‐factor F is C(8)‐linked if, for every i≤k, there is a circuit Di of length 8 with edge sequence e1i…e8i where e1i,e5i∈Efalse(C2i−1false) and e3i,e7i∈Efalse(C2ifalse). 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.

Berge–Fulkerson coloring for C(12)‐linked permutation graphs

“…Since it was proved in [12] that Conjecture 1 holds for the case of ‐linked permutation graphs, we have the following claim (by Proposition 1).…”

“…Among these famous conjectures, the B‐F‐conjecture is less explored than the other two conjectures and is still open for some known snarks. In [5,6,11,12,14], the conjecture is verified for some families of snarks. It was shown in [13], a possible minimum counterexample for the B‐F‐conjecture should have cyclic edge‐connectivity at least 5.…”

“…In [12], the B‐F‐conjecture was verified for all permutation graphs with alternating circuit of length at most 8. This result is further extended to permutation graphs with alternating circuit of length 12 (Theorem 1).…”