Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0907.3705

On hitting all maximum cliques with an independent set

Landon Rabern
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that every graph $G$ for which $ω(G) \geq 3/4(Δ(G) + 1)$, has an independent set $I$ such that $ω(G - I) < ω(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $ω(G) < 3/4(Δ(G) + 1)$ and hence also $χ(G) > \lceil 7/6ω(G) \rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $χ(H) \leq \max{\lceil 7/6ω(H) \rceil, \lceil \frac{ω(H) + Δ(H) + 1}{2}\rceil}$, then we also have $χ(G) \leq \lceil \frac{ω(G) + Δ(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Landon Rabern. 2010-03-12. On hitting all maximum cliques with an independent set. https://arxiv.org/abs/0907.3705

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

KEEP EXPLORING

Related papers

arXiv2026

On Perfect Divisibility of Bull-Free Graphs Without Long Paths

A graph $G$ is {\em perfectly divisible} if, for every induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. Chudnovsky and Sivaraman [J. Graph Theory \textbf{90} (2019) 54-60] proved that every ($P_5$, bull)-free graph is perfectly divisible, while Chen and Xu [Discrete Appl. Math. \textbf{372} (2025) 298-307] proved the same for ($P_7,C_5$, bull)-free graphs. We extend these results by proving that every ($P_8,C_5$, bull)-free graph is perfectly divisible and that, letting $F$ denote the Grötzsch graph, a ($P_6$, bull)-free graph is perfectly divisible if and only if it is $F$-free.

math.CO↗
arXiv2026

Covering graphs by isometric trees

A connected subgraph of a graph is isometric if it preserves distances. Recently, graphs admitting a vertex or edge covering by a small number of isometric paths have been studied. In this paper, we consider the analogous problem for isometric trees, focusing on the treewidth of graphs admitting a vertex or edge covering by a small number of such trees. Baste, De Meyer, Giocanti, Objois, and Picavet showed that for coverings by two isometric trees, the treewidth is bounded. We show that already for three isometric trees, the treewidth can be linear in the number of vertices. On the positive side, we show that for graphs of bounded degree coverable by a small number of isometric trees, the treewidth is sublinear in the number of vertices.

math.CO↗
arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗