Normal edge‐colorings of cubic graphs

Abstract

A normal k‐edge‐coloring of a cubic graph is a proper edge‐coloring with k colors having the additional property that when looking at the set of colors assigned to any edge e and the four edges adjacent to it, we have either exactly five distinct colors or exactly three distinct colors. We denote by χ N true′ ( G ) the smallest k, for which G admits a normal k‐edge‐coloring. Normal k‐edge‐colorings were introduced by Jaeger to study his well‐known Petersen Coloring Conjecture. More precisely, it is known that proving χ N true′ false( G false) ≤ 5 for every bridgeless cubic graph is equivalent to proving the Petersen Coloring Conjecture and then it implies, among others, Cycle Double Cover Conjecture and Berge‐Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with χ N true′ ( G ) = 7. In contrast, the known best general upper bound for χ N true′ ( G ) was 9. Here, we improve it by proving that χ N true′ ( G ) ≤ 7 for any simple cubic graph G, which is best possible. We obtain this result by proving the existence of specific nowhere zero Z 2 2‐flows in 4‐edge‐connected graphs.

Normal 6-edge-colorings of some bridgeless cubic graphs

“…Every nowhere-zero Z 3 2 -flow φ can be seen as a normal 7-edge-coloring of C(n, p) (see Theorem 5 in [17]). Moreover, the first entry of the flow on edges of M is 0, and the first entry of the flow on edges outside M is 1.…”

“…The main result of [17] states that any simple cubic graph admits a normal 7-edgecoloring. There it is also shown that any bridgeless cubic graph G admits a normal 7-edgecoloring (see also [3]), and this result is obtained simply by considering a nowhere zero Z 3 2 -flow of G. One may wonder whether we can choose the nowhere zero Z 3 2 -flow θ of G, such that for one nonzero element γ ∈ Z 3 2 , we have θ −1 (γ) = ∅.…”

“…Let χ ′ N (G) denote the normal chromatic index of G, that is, the minimum number of colors in a normal edge-coloring of G. In this terms, Petersen Coloring Conjecture is equivalent to saying that every bridgeless cubic graph has normal chromatic index at most 5. As far as we know, the best known upper bound for an arbitrary bridgeless cubic graph is 7 (see [3,17]). There exist examples of simple cubic graphs (not bridgeless) with normal chromatic index 7.…”

“…There exist examples of simple cubic graphs (not bridgeless) with normal chromatic index 7. On the other hand, in [17] it is shown that any simple cubic graph admits a normal 7-edge-coloring. Let us recall that a weaker upper bound for an arbitrary simple cubic graph was proved in [3].…”

S_12 and P_12-colorings of cubic graphs

“…Observe that not all cubic graphs admit a normal k-edge-coloring for some k. An example of such a graph is the graph from Figure 2. On the positive side, all simple cubic graphs admit a normal 7edge-coloring [10]. The smallest k (if it exists) for which a cubic graph G admits a normal k-edge-coloring is called a normal chromatic index of G and is denoted by χ N (G).…”