Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring.

💡 Research Summary

The paper addresses the problem of linear list‑coloring in subcubic graphs, that is, graphs whose maximum degree Δ does not exceed three. A linear coloring is a proper vertex coloring with the additional requirement that any two color classes together induce a subgraph that is a disjoint union of paths. The authors prove that for every connected subcubic graph G, and for any assignment of a list L(v) of four admissible colors to each vertex v, there exists a linear coloring φ such that φ(v)∈L(v) for all v and, moreover, the two neighbors of every degree‑2 vertex receive distinct colors. The only graphs that fail to satisfy this property are the 5‑cycle C₅ and the complete bipartite graph K₍₃,₃₎. This result settles a conjecture posed by Esperet, Montassier, and Raspaud.

The proof follows the classic minimal counterexample strategy. Assuming a smallest counterexample G, the authors first show that G must be 2‑connected; otherwise the problem could be solved independently on its blocks and combined. They then identify a collection of reducible configurations—structures that can be safely removed or contracted while preserving the list‑coloring constraints. These include pendant vertices, degree‑2 vertices whose two neighbors are non‑adjacent, and certain arrangements of degree‑3 vertices adjacent to degree‑2 vertices. By repeatedly eliminating such configurations, the graph is reduced to a “core” that contains no reducible patterns.

For this core the authors employ a discharging argument. Each vertex v is given an initial charge μ(v)=deg(v)−2, so degree‑2 vertices start with charge 0 and degree‑3 vertices with charge 1. Discharging rules transfer a fraction of charge from degree‑3 vertices to adjacent degree‑2 vertices in a way that respects the structural restrictions derived from the reducible‑configuration analysis. After discharging, every vertex ends with non‑negative charge, yet the total initial charge of a connected graph equals Σ(deg(v)−2)=2|E|−2|V|=−2, a contradiction. Hence no minimal counterexample can exist, establishing the theorem.

The constructive nature of the proof yields an explicit linear‑time algorithm. The algorithm proceeds in three phases: (1) detection and removal of reducible configurations using adjacency lists, (2) a linear‑time discharging phase that assigns provisional colors to the remaining vertices, and (3) a back‑tracking reconstruction that restores the removed vertices while respecting their lists. Each phase touches each vertex a constant number of times, guaranteeing O(n) time and O(n) space for a graph with n vertices.

Beyond confirming the conjecture, the paper contributes several methodological advances. It demonstrates how list constraints can be integrated into classic reducibility and discharging frameworks, and it shows that the additional requirement on degree‑2 vertices (their neighbors must be differently colored) does not increase the needed list size. The authors also discuss extensions, suggesting that similar techniques might resolve the analogous problem for graphs of maximum degree four with lists of size five, and they point out potential applications in network routing where path‑wise color separation is desirable.

In summary, the authors prove that every connected subcubic graph, except C₅ and K₍₃,₃₎, admits a linear coloring from arbitrary 4‑element lists, and they provide a linear‑time algorithm to construct such a coloring. This resolves the Esperet‑Montassier‑Raspaud conjecture and opens new avenues for research on list‑coloring under structural constraints.