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Hadwiger's conjecture and topological bounds

Raphael Steiner

TL;DR

This work connects topological lower bounds on graph chromatic number to minor and odd-minor containment, advancing the search for (counter)examples to Hadwiger-type conjectures. It shows that a strong combinatorial parameter, the zig-zag-number, enforces large odd cliques as odd-minors whenever it is large, answering a long-standing question by Simonyi and Zsbán. It also sharpens understanding of the Dol'nikov-Kříž bound by proving that large values imply traditional minors, and it establishes the Odd Hadwiger conjecture for Kneser and Schrijver graphs, where topological bounds are tight. Overall, the paper bridges topology-inspired lower bounds with concrete minor- and odd-minor-structural consequences, highlighting the reach and limits of topological methods in Hadwiger-type problems.

Abstract

The Odd Hadwiger's conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger's famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic number, introduced by Matoušek and Ziegler (2003) and refined recently by Daneshpajouh and Meunier (2023), forms a potential avenue to a disproof of Hadwiger's conjecture or its odd-minor variant. In this direction, we prove that, in a very general sense, every graph $G$ that admits a topological lower bound of $t$ on its chromatic number, contains $K_{\lfloor t/2\rfloor +1}$ as an odd-minor. This solves a problem posed by Simonyi and Zsbán [European Journal of Combinatorics, 31(8), 2110--2119 (2010)]. We also prove that if for a graph $G$ the Dol'nikov-Kříž lower bound on the chromatic number (one of the lower bounds in the aforementioned hierarchy) attains a value of at least $t$, then $G$ contains $K_t$ as a minor. Finally, extending results by Simonyi and Zsbán, we show that the Odd Hadwiger's conjecture holds for Schrijver and Kneser graphs for any choice of the parameters. The latter are canonical examples of graphs for which topological lower bounds on the chromatic number are tight.

Hadwiger's conjecture and topological bounds

TL;DR

This work connects topological lower bounds on graph chromatic number to minor and odd-minor containment, advancing the search for (counter)examples to Hadwiger-type conjectures. It shows that a strong combinatorial parameter, the zig-zag-number, enforces large odd cliques as odd-minors whenever it is large, answering a long-standing question by Simonyi and Zsbán. It also sharpens understanding of the Dol'nikov-Kříž bound by proving that large values imply traditional minors, and it establishes the Odd Hadwiger conjecture for Kneser and Schrijver graphs, where topological bounds are tight. Overall, the paper bridges topology-inspired lower bounds with concrete minor- and odd-minor-structural consequences, highlighting the reach and limits of topological methods in Hadwiger-type problems.

Abstract

The Odd Hadwiger's conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger's famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic number, introduced by Matoušek and Ziegler (2003) and refined recently by Daneshpajouh and Meunier (2023), forms a potential avenue to a disproof of Hadwiger's conjecture or its odd-minor variant. In this direction, we prove that, in a very general sense, every graph that admits a topological lower bound of on its chromatic number, contains as an odd-minor. This solves a problem posed by Simonyi and Zsbán [European Journal of Combinatorics, 31(8), 2110--2119 (2010)]. We also prove that if for a graph the Dol'nikov-Kříž lower bound on the chromatic number (one of the lower bounds in the aforementioned hierarchy) attains a value of at least , then contains as a minor. Finally, extending results by Simonyi and Zsbán, we show that the Odd Hadwiger's conjecture holds for Schrijver and Kneser graphs for any choice of the parameters. The latter are canonical examples of graphs for which topological lower bounds on the chromatic number are tight.
Paper Structure (11 sections, 5 theorems, 7 equations, 2 figures)

This paper contains 11 sections, 5 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Topological lower bounds.
  3. Combinatorial parameters.
  4. Our results.
  5. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:dolnikov']}
  6. Claim ($\ast$). Let $I \subseteq [n]$ and let $f:\bigcup_{i \in I}{V(T_i)}\rightarrow \{1,2\}$ be any assignment that, restricted to each tree $T_i$ with $i \in I$, forms a proper coloring. If for some distinct indices $i, j \in I$ there exists an edge in $G$ connecting $T_i$ and $T_j$, then there also exists a monochromatic edge (with respect to $f$) connecting $T_i$ and $T_j$.
  7. Case 1. $|M|\le k-2$.
  8. Case 2. $|M|=k-1$.
  9. Claim 1. For every $u \in U$ and every $i \in \{1,2\}$ there exists a hyperedge $e \in E$ such that $e \subseteq X_i \cup \{u\}$ and $u \in e$.
  10. Claim 2. For every pair of integers $n_1, n_2\ge 0$ such that $n_1+n_2\ge t$, the graph $K_{n_1}+K_{n_2,n_2}^\ast$ contains $K_t$ as a minor.
  11. Odd Hadwiger's conjecture for Kneser and Schrijver graphs

Key Result

Theorem 1

Let $G$ be a graph with a Kneser-representation $\mathcal{H}$. Then $\chi(G)\ge \mathrm{cd}(\mathcal{H})$.

Figures (2)

  • Figure 1: Illustration of the topological lower bounds on the chromatic number of a graph and their relations.
  • Figure 2: Illustration for the construction of the odd $K_{n-2k+2}$-expansion in $S(n,k)$ for the case $n=14, k=4$. Colors red and blue in the picture correspond to colors $1$ and $2$ in the proof, respectively. For the sake of readability, only some of the monochromatic edges connecting pairs of distinct trees $T_i$ and $T_j$ are indicated.

Theorems & Definitions (18)

  • Conjecture 1: Hadwiger 1943 hadwiger
  • Conjecture 2: Gerards and Seymour 1995 gerards
  • Definition 1: Box complexes, cf. matousekzieglerdanesh
  • Definition 2: Zig-Zag-number, cf. danesh, Section 4.1
  • Definition 3: $2$-colorability defect, Kneser representation, cf. matousekziegler, page 8--9 and danesh, Section 4.1
  • Theorem 1: Dol'nikov dolnikov, Kříž kriz
  • Theorem 2
  • Theorem 3
  • Definition 4: Kneser graph, Schrijver graph, cf. schrijver
  • Theorem 4
  • ...and 8 more