On 3-colorability of (claw, diamond)-free graphs

TL;DR

The paper studies $3$-colorability in classes of claw- and diamond-free graphs augmented by forbidding generalised nets $N_{i,j,k}$. By reducing $3$-coloring to $3$-edge-coloring on an underlying subcubic triangle-free graph $H$ via line-graph representations, the authors derive structural dichotomies: graphs are either $3$-colorable, contain a $K_4$, or belong to explicitly described exceptional families (e.g., $B_{10}$, $B_{9i+1}$, $B_{9i+7}$, and necklace-based constructions $D_{6i+1}$, $D_{6i+5}$). These results yield finite and infinite families of non-$3$-colorable examples and, in several cases, imply polynomial-time decision procedures for $3$-colorability within the restricted classes. The work extends the understanding of the boundary between NP-complete and tractable coloring problems under forbidden induced subgraphs, and connects line-graph theory with edge-coloring in subcubic triangle-free graphs, offering concrete exceptional structures and constructive reductions.

Abstract

The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $(claw, diamond, K_4)$-free graphs. Recently, $3$-colorability has been also considered for $(claw, N_{1,1,1})$-free graphs. Here, a generalised net $N_{i, j, k}$ is the graph obtained by identifying each vertex of a triangle with an endvertex of one of three vetex-disjont paths of lengths $i, j, k$. We study the class of $(claw, diamond, N_{i, j, k})$-free graphs for $(i, j, k) \in \{(1, 1, 3),$ $ (1, 2, 2), $ $(2, 2, 2) \}$. We show that these graphs are $3$-colorable or contain a $K_4$ or belong to some well-defined class of non $3$-colorable graphs. Moreover, we prove that there are only finitely many non $3$-colorable $N_{1, 2, k}$-free graphs for any $k \geq 2$, but there exist infinitely many non $3$-colorable $N_{i, j, k}$-free graphs for any $2 \leq i \leq j \leq k.$

Figures (11)

• Figure 1: The graphs called the bull, the net and $N_{1,2,3}$ respectively.
• Figure 2: The exceptional non-$3$-colorable, $(claw, diamond)$-free graphs and their base graphs.
• Figure 3: Forbidden subgraph in Case 1 in the proof of Theorem \ref{['thm::N113']}.
• Figure 4: Forbidden subgraphs in Case 2 of Theorem \ref{['thm::N113']}.
• Figure 5: Forbidden subgraphs and coloring of neighborhood of $C_5$ in Case 2 of Theorem \ref{['thm::N113']}.

Theorems & Definitions (25)

• Theorem 1
• Corollary 2
• Theorem 4
• proof
• Theorem 7
• proof
• Theorem 8
• Theorem 9
• proof
• Lemma 10