List strong and list normal edge-coloring of (sub)cubic graphs

TL;DR

The paper investigates both strong and normal edge-colorings and their list analogues in (sub)cubic graphs. It proves a tight bound $\chi_{s,l}'(G) \le 10$ for subcubic graphs and constructs infinite families where the list strong chromatic index exceeds the ordinary strong index, including a precise result $\chi_{s,l}'(P)=7$ for the Petersen graph. It also initiates the study of list normal edge-coloring, showing substantial gaps between $\chi_n'(G)$ and $\chi_{n,l}'(G)$ with lower bounds reaching 9 in some cubic graphs and 7 in cyclically 4-edge-connected cases. Together, these results advance understanding of how list constraints affect edge-coloring in low-degree graphs and connect to Petersen-related coloring conjectures and related combinatorial techniques.

Abstract

A strong edge-coloring of a graph is a proper edge-coloring, in which the edges of every path of length 3 receive distinct colors; in other words, every pair of edges at distance at most 2 must be colored differently. The least number of colors needed for a strong edge-coloring of a graph is the strong chromatic index. We consider the list version of the coloring and prove that the list strong chromatic index of graphs with maximum degree 3 is at most 10. This bound is tight and improves the previous bound of 11 colors. We also consider the question whether the strong chromatic index and the list strong chromatic index always coincide. We answer it in negative by presenting an infinite family of graphs for which the two invariants differ. For the special case of the Petersen graph, we show that its list strong chromatic index equals 7, while its strong chromatic index is 5. Up to our best knowledge, this is the first known edge-coloring for which there are graphs with distinct values of the chromatic index and its list version. In relation to the above, we also initiate the study of the list version of the normal edge-coloring. A normal edge-coloring of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with 4 colors or to edges colored with 2 colors. It is conjectured that 5 colors suffice for a normal edge-coloring of any bridgeless cubic graph which is equivalent to the Petersen Coloring Conjecture. Similarly to strong edge-coloring, list normal edge-coloring is much more restrictive and consequently for many graphs the list normal chromatic index is greater than the normal chromatic index. We show that there are cubic graphs with list normal chromatic index at least $9$, there are bridgeless cubic graphs with its value at least 8, and there are cyclically 4-edge-connected cubic graphs with value at least 7.

Figures (10)

• Figure 1: The hypothetical $6$-cycle $C$ in $G$. The edges $y_1$ and $y_4$ (and also $y_2$ and $y_5$) might be connected by an edge (depicted dashed).
• Figure 2: The hypothetical $k$-cycle $C$ in $G$.
• Figure 3: The graph $P'$ obtained from the Petersen graph by replacing one edge with two pendant edges.
• Figure 4: The dodecahedron (left) and the generalized Petersen graph $GP(10,3)$ (right).
• Figure 5: The Petersen graph $P$.

Theorems & Definitions (31)

• Conjecture 1.1: Erdős, Nešetřil Erd88
• Theorem 1.2: Andersen And92; Horák, Qing, and Trotter HorQinTro93
• Conjecture 1.3: Lužar, Máčajová, Škoviera, and Soták LuzMacSkoSot22
• Theorem 1.4
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.8: Jaeger Jae85
• Corollary 1.9
• Theorem 2.1
• Theorem 2.2: Combinatorial Nullstellensatz Alo99