Immersions of large cliques in graphs with independence number 2 and bounded maximum degree

TL;DR

This paper proves that graphs with independence number 2, bounded maximum degree, and limited clique covering contain immersions of complete graphs matching their chromatic number, advancing understanding of the immersion analogue of Hadwiger's conjecture.

Contribution

It verifies Vergara's conjecture for graphs with independence number 2 under certain maximum degree and clique covering constraints, extending previous results.

Findings

Graphs with independence number 2 and bounded maximum degree contain large clique immersions.

The result applies to graphs with maximum degree less than 2n/3 - 1 and clique covering number at most 3.

Improves bounds on maximum degree for the existence of complete graph immersions in such graphs.

Abstract

An immersion of a graph $H$ in a graph $G$ is a minimal subgraph $I$ of $G$ for which there is an injection ${{\rm i}} \colon V(H) \to V(I)$ and a set of edge-disjoint paths $\{P_e: e \in E(H)\}$ in $I$ such that the end vertices of $P_{uv}$ are precisely ${{\rm i}}(u)$ and ${{\rm i}}(v)$. The immersion analogue of Hadwiger Conjecture (1943), posed by Lescure and Meyniel (1985), asks whether every graph $G$ contains an immersion of $K_{\chi(G)}$. Its restriction to graphs with independence number 2 has received some attention recently, and Vergara (2017) raised the weaker conjecture that every graph with independence number 2 has an immersion of $K_{\chi(G)}$. This implies that every graph with independence number 2 has an immersion of $K_{\lceil n/2 \rceil}$. In this paper, we verify Vergara Conjecture for graphs with bounded maximum degree. Specifically, we prove that if $G$ is a graph with independence number $2$, maximum degree less than $2n/3 - 1$ and clique covering number at most $3$, then $G$ contains an immersion of $K_{\chi(G)}$ (and thus of $K_{\lceil n/2 \rceil}$). Using a result of Jin (1995), this implies that if $G$ is a graph with independence number $2$ and maximum degree less than $19n/29 - 1$, then $G$ contains an immersion of $K_{\chi(G)}$ (and thus of $K_{\lceil n/2 \rceil}$).