Fulkerson-covers of hypohamiltonian graphs

Covering a cubic graph by 5 perfect matchings

“…A graph G is called hypohamiltonian if G itself is not hamiltonian but the removel of any vertex of G results a hamiltonian graph. Chen and Fan [1] verified the Fulkerson Conjecture for several known classes of hypohamiltonian graphs in the literatures. Now Höggkvist's conjecture is still open.…”

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

“…This revision eliminates the introduction of a new term “vertexless loop” that used in . (Furthermore, a generalized version of the lemma using the term of 4‐flow can be applied in the study of 6‐cycle 4‐covering problem for noncubic graphs.) Definition Let G be a graph.…”

“…The original version of Lemma in uses the term of 3‐edge‐coloring instead of 4‐flow. Using the term 4‐flow in the lemma was suggested in since it was pointed out by Tutte that the admission of a nowhere‐zero 4‐flow is equivalent to the 3‐edge‐coloring for cubic graphs. This revision eliminates the introduction of a new term “vertexless loop” that used in .…”

“…Some of these families were never studied before for the conjecture. Some of them were already verified in some literatures (such as, in , , etc.). However, by applying Corollary , new proofs are simpler.…”

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

“…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.…”