Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.00202

On graphs without four-vertex induced subgraphs

Kathie Cameron·Chính T. Hoàng·Taite LaGrange
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a family F of graphs, a graph G is F-free if it does not contain any graph in F as an induced subgraph. The problem of determining the complexity of colouring (claw, 4K1)- free graphs is a well-known open problem. In this paper we solve the colouring problem for a subclass of (claw, 4K1)-free graphs. We design a polynomial-time algorithm to colour (claw, 4K1, bridge, C4-twin)-free graphs. This algorithm is derived from a structural theorem on (claw, 4K1, bridge, C4-twin)-free graphs.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kathie Cameron, Chính T. Hoàng, Taite LaGrange. 2025-04-30. On graphs without four-vertex induced subgraphs. https://arxiv.org/abs/2505.00202

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗