Berge–Fulkerson Conjecture on Certain Snarks

Normal 5-edge-colorings of a family of Loupekhine snarks

“…Related with this question, in this paper we show that a class of snarks obtained with a method introduced by Loupekhine, admits a normal edgecoloring with five colors. Let us remark that this result implies the main result in [11]. Now, let us give all necessary definitions and notions used in the paper.…”

“…Now we construct a normal 5-edge-colorings for a class of LP 1 -graphs. Later, we will prove that our class properly contains all graphs considered in [11].…”

“…If the k blocks can be partitioned in three subsequences of odd length in such a way that properties P1 and P2 hold, then G L k admits a normal 5-edge-coloring. Now, let us show that the class of LP 1 -graphs presented in [11] satisfy the assumptions of Theorem 1.…”

“…In [11], the authors prove that a very special family of LP 1 -graphs admits a Berge-Fulkerson covering. In the next section, we strongly generalize their result by proving the existence of a normal 5-edge-coloring for a larger class of LP 1 -graphs which properly contains the class of LP 1 -graphs studied in [11]. Hence, by Proposition 2, we have that all of them admit also a Berge-Fulkerson covering.…”

“…For any odd k > 1, the LP 1 -graph considered in [11] is the one having the following decomposition in G C k and G B k : The subgraph G C k has a unique star and denote its three vertices of degree one by z 1 , z bk=2cþ1 and z bk=2cþ2 : Moreover, each copy of K 2 connects pairwise the vertices z iþ2 , z kÀi , for i ¼ 0, :::, bk=2c À 2:…”