On 3-Coloring of $(2P_4,C_5)$-Free Graphs

TL;DR

This paper resolves the polynomial-time solvability of 3-coloring for the hereditary class of $(2P_4,C_5)$-free graphs, addressing a long-standing gap in the complexity landscape for small forbidden subgraph sets. The authors reduce the problem to list-3-coloring with a bounded initial subgraph $N_0$, then apply a suite of reductions (including diamond consistency, cut reductions, and neighborhood collapse) and strategic branching to obtain a polynomial number of simpler instances. Most of these instances are encoded as $2$-SAT, enabling efficient resolution, while the analysis distinguishes between $C_7$-free and $C_7$-containing graphs and further refines the structure around induced cycles $C_7$ and $C_9$. The result advances our understanding of 3-coloring in restricted graph classes and outlines a roadmap toward resolving the broader open cases for $2P_4$-free graphs, with potential extensions to list-3-coloring.

Abstract

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs in the class are called $(H_1,H_2,\ldots)$-free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For $H$-free graphs, the complexity is settled for any $H$ on up to seven vertices. There are only two unsolved cases on eight vertices, namely $2P_4$ and $P_8$. For $P_8$-free graphs, some partial results are known, but to the best of our knowledge, $2P_4$-free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $(2P_4,C_5)$-free graphs.

Figures (13)

• Figure 1: Picture showing the induced $2P_4$ in the case of $G$ being $C_7$-free. If the dash-and-dotted edge is present, $A$ has length 7, otherwise $A$ has length 9.
• Figure 2: Finding an induced $2P_4$, assuming $P$ is an induced $P_4$ with exactly three vertices in $N_2$. Note that $P$ can look differently, but always contains $x$.
• Figure 3: Vertex $x \in N_1$ being a partial neighbor of a top component $C$ and neighboring another top component $C'$ leads to an induced $2P_4$.
• Figure 4: Illustrations of the situations in the proof of Lemma \ref{['lem-Ri']}.
• Figure 5: Considering a pair of vertices $(x,y)$ of type $\gamma$ for $x \in S_i, y \in S_{i+3}$, the edge $x'y'$ must be present, otherwise we obtain an induced $2P_4$. The other dash-and-dotted edges are not necessarily present, and the vertex $v_{i+5}$ is adjacent to at most one vertex from $\{x, y\}$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2: GJPS16
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof