Non-conflicting no-where zero $Z_2\times Z_2$ flows in cubic graphs

TL;DR

Problem: determine whether bridgeless cubic graphs admit normal $6$-edge-colorings via non-conflicting $Z_2\\times Z_2$-flows on $G/\\overline{F}$ for some perfect matching $F$. Approach: define non-conflicting flows and prove existence in claw-free bridgeless cubic graphs and in graphs with a $2$-factor with at most two cycles; connect these flows to Thomassen’s conjecture on edge-disjoint perfect matchings and explore implications for normal chromatic index. Key results: (i) claw-free bridgeless cubic graphs have an $F$ with the desired flow; (ii) such flows exist for graphs with at most two odd cycles in the 2-factor, excluding the Petersen graph; (iii) there are infinitely many $2$-edge-connected cubic graphs without such flows for any $F$; (iv) a flow-based path to $\ hoambda_N'(G)\le 6$ in relevant classes. Significance: advances toward the Petersen Coloring Conjecture and the six-edge-coloring threshold for bridgeless cubic graphs, while clarifying limits via explicit counterexamples and linking flow concepts to broader conjectures.

Abstract

Let $Z_2\times Z_2=\{0, α, β, α+β\}$. If $G$ is a bridgeless cubic graph, $F$ is a perfect matching of $G$ and $\overline{F}$ is the complementary 2-factor of $F$, then a no-where zero $Z_2\times Z_2$-flow $θ$ of $G/\overline{F}$ is called non-conflicting with respect to $\overline{F}$, if $\overline{F}$ contains no edge $e=uv$, such that $u$ is incident to an edge with $θ$-value $α$ and $v$ is incident to an edge with $θ$-value $β$. In this paper, we demonstrate the usefulness of non-conflicting flows by showing that if a cubic graph $G$ admits such a flow with respect to some perfect matching $F$, then $G$ admits a normal 6-edge-coloring. We use this observation in order to show that claw-free bridgeless cubic graphs, bridgeless cubic graphs possessing a 2-factor having at most two cycles admit a normal 6-edge-coloring. We demonstrate the usefulness of non-conflicting flows further by relating them to a recent conjecture of Thomassen about edge-disjoint perfect matchings in highly connected regular graphs. In the end of the paper, we construct infinitely many 2-edge-connected cubic graphs such that $G/\overline{F}$ does not admit a non-conflicting no-where zero $Z_2\times Z_2$-flow with respect to any perfect matching $F$.

Figures (9)

• Figure 1: An example of an $H$-coloring of $G$.
• Figure 2: The graph $P_{10}$.
• Figure 3: A cubic graph that requires $7$ colors in a normal coloring. The bridge is poor. All other edges are rich. It can be shown that $\chi'_N(G)=7$.
• Figure 4: A cubic graph that does not admit a normal $k$-edge-coloring for any $k\geq 1$.
• Figure 5: The graph $K_4$.

Theorems & Definitions (11)

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