[Paper Review] A Note on (3,1)-Choosable Toroidal Graphs

Baogang Xu, Qinglin Yu|ArXiv.org|Sep 27, 2006

Graph Labeling and Dimension Problems4 references3 citations

TL;DR

This paper proves that every toroidal graph without adjacent triangles, 6-cycles, and cycles of length 5 or 7 is (3,1)*-choosable, meaning it admits a list coloring where each vertex selects a color from a list of size 3, with at most one neighbor sharing its color. Using discharging techniques and structural analysis, the authors identify forbidden configurations and show that such graphs always allow a valid (3,1)*-coloring under the given constraints.

ABSTRACT

An $(L,d)^*$-coloring is a mapping $ϕ$ that assigns a color $ϕ(v)\in L(v)$ to each vertex $v\in V(G)$ such that at most $d$ neighbors of $v$ receive colore $ϕ(v)$. A graph is called $(m,d)^*$-choosable, if $G$ admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\geq m$ for all $v\in V(G)$. In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles and $l$-cycles for some $l \in \{5,7\}$, is $(3,1)^*$-choosable.

Motivation & Objective

• To determine sufficient conditions under which toroidal graphs are (3,1)*-choosable.
• To extend prior results on (4,1)*-choosability of triangle-free toroidal graphs to the tighter (3,1)*-choosability condition.
• To identify structural obstructions (forbidden configurations) that prevent (3,1)*-choosability in toroidal graphs with specific cycle and face constraints.
• To resolve the open problem of (3,1)*-choosability for triangle-free toroidal graphs with restricted cycle lengths.

Proposed method

• Employing the discharging method to redistribute initial charges based on vertex and face degrees, assigning initial charge ω(v) = d(v)/3 − 1 and ω(f) = d(f)/6 − 1.
• Defining discharging rules that transfer charge from vertices to adjacent faces, particularly favoring higher-degree faces and vertices of degree 3 or 4.
• Analyzing the final charge distribution to derive contradictions under the assumption that none of the forbidden configurations (δ(G) < 3, adjacent 3-vertices, (3,4,4)-faces, (3,4,3,4)-faces) exist.
• Using extremal graph theory: assuming a minimal counterexample G with minimum number of vertices to prove (3,1)*-choosability by induction on subgraphs.
• Applying case analysis based on face degrees (3, 4, 5, ≥8) and vertex degrees to bound charge transfer and ensure non-negative final charges.
• Verifying that all face and vertex types satisfy ω′(x) ≥ 0 under the discharging rules, leading to a contradiction with Euler’s formula.

Experimental results

Research questions

• RQ1 Under what structural conditions is a toroidal graph (3,1)*-choosable?
• RQ2 Can the (4,1)*-choosability result for triangle-free toroidal graphs be strengthened to (3,1)*-choosability?
• RQ3 What forbidden configurations must be absent in a toroidal graph to ensure (3,1)*-choosability?
• RQ4 Is the absence of 6-cycles and cycles of length 5 or 7 sufficient for (3,1)*-choosability in triangle-free toroidal graphs?

Key findings

• Every toroidal graph containing no adjacent triangles, no 6-cycles, and no cycles of length 5 or 7 is (3,1)*-choosable.
• The absence of adjacent 3-vertices, (3,4,4)-faces, and (3,4,3,4)-faces is necessary for the (3,1)*-choosability result.
• The discharging method successfully identifies and eliminates all potential minimal counterexamples by showing their charge sum cannot be zero.
• The proof relies on the fact that any minimal counterexample must contain one of four forbidden configurations, all of which are ruled out by the graph constraints.
• The result improves upon prior work by Xu and Zhang, who showed (4,1)*-choosability for triangle-free toroidal graphs, by reducing the list size requirement from 4 to 3.