Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917013155217408 |
|---|---|
| author | Song, Zi-Xia Tibbetts, Thomas |
| author_facet | Song, Zi-Xia Tibbetts, Thomas |
| contents | A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing ``every vertex in $T_j$'' by ``some vertex in $T_j$'' retrieves the standard definition of a $K_t$ minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This is a substantial strengthening of the celebrated Hadwiger's Conjecture, which asserts that every graph with chromatic number $t$ contains a $K_t$ minor. At the ``New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we prove that the Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs Song, Zi-Xia Tibbetts, Thomas Combinatorics A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing ``every vertex in $T_j$'' by ``some vertex in $T_j$'' retrieves the standard definition of a $K_t$ minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This is a substantial strengthening of the celebrated Hadwiger's Conjecture, which asserts that every graph with chromatic number $t$ contains a $K_t$ minor. At the ``New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we prove that the Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four. |
| title | Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.12567 |