Biclique immersions in graphs with independence number 2

TL;DR

This work studies the immersion analogue of Hadwiger's conjecture for graphs with independence number $2$, proving that every $n$-vertex graph with $\alpha(G)=2$ contains an immersion of every complete bipartite graph on $\lceil n/2\rceil$ vertices, i.e., $K_{\ell,\lceil n/2\rceil-\ell}$ for $\ell\le \lceil n/2\rceil-1$. The authors implement a careful inductive proof on $n+\ell$ centered on edge-critical reductions, a partition of the vertex set into $C$, $X$, and $Y$, and a key matching-lemma derived from König’s line coloring to ensure edge-disjoint paths realize the immersion. They extend these immersions to yield corollaries for chromatic-number-related immersions, notably $K_{\ell,\chi(G)-\ell}$ for $1\le \ell \le \chi(G)-1$, and prove a strengthening $K_{1,1,\chi(G)-2}$ when $\chi(G)\ge 3$, thus tying immersion results to broader conjectures on dense substructures. Additionally, a separate result on edge-disjoint immersions of matchings in complete graphs is provided, accompanied by open questions for further generalization. Overall, the paper advances the theory of graph immersions in low-independence-number settings and links biclique immersions to chromatic parameters with potential impact on related immersion-minor theory.

Abstract

The analogue of Hadwiger's conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{χ(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n/2 \rceil}$. We show that every $n$-vertex graph with independence number 2 contains every complete bipartite graph on $\lceil n/2 \rceil$ vertices as an immersion.

Figures (11)

• Figure 1: An example of Lemma \ref{['lem:matching_lemma']}.
• Figure 2: \ref{['fig:induction_immersion']} Partition of $V(G)$ with respect to $H'$; \ref{['fig:paths_using_Oc']} Paths from $x$ to $L_y$ using $O_c$ and $y$ depicted in dashed lines.
• Figure 3: Partition of $G$ according to non-adjacent vertices $x$ and $y$ and their common neighborhood $C$.
• Figure 4: Immersion for Case 1, when $|\overline{X}_C|, |\overline{Y}_C| < \ell$: between $Y^* \cup \overline{X}_a$ and $X^* \cup \overline{Y}_a$.
• Figure 5: Construction for Case 1, connecting $X^*$ to $Y^*$ through $C$.

Theorems & Definitions (28)

• Conjecture 1: Lescure and Meyniel LescureM1989
• Conjecture 2: Vergara Vergara2017
• Theorem 3
• Corollary 4
• Conjecture 5
• Proposition 6
• Conjecture 7
• Lemma 8
• proof
• Claim 9