Immersions and Albertson's conjecture

Abstract

A graph is said to contain $K_k$ (a clique of size $k$) as a weak immersion if it has $k$ vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger's conjecture: Every graph of chromatic number $k$ contains $K_k$ as a weak immersion. We prove this conjecture for graphs with at most $(1.64-o(1))k$ vertices. As an application, we make some progress on Albertson's conjecture, according to which every graph $G$ with chromatic number $k$ satisfies $cr(G) \geq cr(K_k)$. In particular, we show that the conjecture is true for all graphs of chromatic number $k$, provided that they have at most $(1.64-o(1))k$ vertices.

Figures (1)

• Figure 1: Constructing a path from $u$ to $u'$.

Theorems & Definitions (31)

• Conjecture 1.1: LM
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 2.1: Ga
• Lemma 2.2
• proof : Proof of Theorem \ref{['main1']}(i)
• Lemma 2.3
• proof : Proof of Theorem \ref{['main1']}(ii)
• Lemma 3.1