[Paper Review] Circular chromatic index of graphs of maximum degree 3

Peyman Afshani, Ghandehari Mahsa+4 more|arXiv (Cornell University)|Dec 31, 2006

Retinoids in leukemia and cellular processes7 references17 citations

TL;DR

This paper proves that for any graph $G$ of maximum degree 3 that does not contain $H_1$ or $H_2$ as a subgraph—where $H_1$ and $H_2$ are specific graphs derived from $K_2^3$ and $K_4$—the circular chromatic index $\chi_c'(G) \leq 11/3$. The result implies no graph exists with $11/3 < \chi_c'(G) < 4$, resolving a key question about gaps in circular chromatic indices and confirming a conjecture for 2-edge-connected cubic graphs.

ABSTRACT

This paper proves that if $G$ is a graph (parallel edges allowed) of maximum degree 3, then $χ_c'(G) \leq 11/3$ provided that $G$ does not contain $H_1$ or $H_2$ as a subgraph, where $H_1$ and $H_2$ are obtained by subdividing one edge of $K_2^3$ (the graph with three parallel edges between two vertices) and $K_4$, respectively. As $χ_c'(H_1) = χ_c'(H_2) = 4$, our result implies that there is no graph $G$ with $11/3 &lt; χ_c'(G) &lt; 4$. It also implies that if $G$ is a 2-edge connected cubic graph, then $χ'(G) \le 11/3$.

Motivation & Objective

• To determine the exact upper bound for the circular chromatic index of graphs with maximum degree 3 that avoid specific subgraphs.
• To resolve whether there exist graphs with circular chromatic index in the open interval $ (11/3, 4) $.
• To prove that all 2-edge-connected cubic graphs have circular chromatic index at most $ 11/3 $, supporting the Petersen Coloring Conjecture.

Proposed method

• Uses induction on the number of edges, reducing the problem by contracting triangles or removing parallel edges.
• Applies the concept of $(p,q)$-colorings and the directed graph $D_c(G)$ of tight arcs to bound the circular chromatic number.
• Employs a strengthened version of a known lemma: if $D_c(G)$ is acyclic and each path contains at most $n$ vertices of color $k-1$, then $\chi_c(G) \leq k - \frac{1}{n+1}$.
• Analyzes line graphs $L(G)$ and constructs $(11,3)$-colorings via edge contraction and extension techniques.
• Considers two cases: graphs with parallel edges (excluding $H_1$) and graphs with triangles (excluding $H_2$), using structural graph theory.
• Provides explicit $(7,2)$-colorings for critical cases where contraction yields $H_1$ or $H_2$, confirming $\chi_c'(G) = 4$ for these graphs.

Experimental results

Research questions

• RQ1 Is there any graph $G$ with $11/3 < \chi_c'(G) < 4$?
• RQ2 What is the maximum possible circular chromatic index for graphs of maximum degree 3 that avoid $H_1$ and $H_2$?
• RQ3 Do all 2-edge-connected cubic graphs satisfy $\chi_c'(G) \leq 11/3$?
• RQ4 Can the Petersen Coloring Conjecture be supported via circular chromatic index bounds?
• RQ5 Are there gaps in the set of realizable circular chromatic indices for graphs, and if so, what are their sizes?

Key findings

• The circular chromatic index of any graph $G$ with maximum degree 3 that does not contain $H_1$ or $H_2$ as a subgraph is at most $11/3$.
• There is no graph $G$ with $11/3 < \chi_c'(G) < 4$, thus closing the interval $ (11/3, 4) $ as a gap in the set of possible circular chromatic indices.
• All 2-edge-connected cubic graphs satisfy $\chi_c'(G) \leq 11/3$, confirming a key implication of the Petersen Coloring Conjecture.
• The graphs $H_1$ and $H_2$ are the only graphs of maximum degree 3 with $\chi_c'(G) = 4$, and they are characterized by their forbidden subgraph structure.
• For triangle-free cubic graphs of girth at least 4, the circular chromatic index is at most $11/3$, proven via perfect matching extension and coloring extension techniques.
• The paper provides explicit $(7,2)$-colorings for graphs that contract to $H_1$ or $H_2$, confirming their circular chromatic index is exactly 4.