Dominating $K_t$-Models
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929354768908288 |
|---|---|
| author | Illingworth, Freddie Wood, David R. |
| author_facet | Illingworth, Freddie Wood, David R. |
| contents | A $\textit{dominating $K_t$-model}$ in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leqslant i<j \leqslant t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to $K_t$ being a minor of $G$. We explore in what sense dominating $K_t$-models behave like (non-dominating) $K_t$-models. The two notions are equivalent for $t \leqslant 3$, but are already very different for $t = 4$, since the 1-subdivision of any graph has no dominating $K_4$-model. Nevertheless, we show that every graph with no dominating $K_4$-model is 2-degenerate and 3-colourable. More generally, we prove that every graph with no dominating $K_t$-model is $2^{t-2}$-colourable.
Motivated by the connection to chromatic number, we study the maximum average degree of graphs with no dominating $K_t$-model. We give an upper bound of $2^{t-2}$, and show that random graphs provide a lower bound of $(1-o(1))t\log t$, which we conjecture is asymptotically tight. This result is in contrast to the $K_t$-minor-free setting, where the maximum average degree is $Θ(t\sqrt{\log t})$. The natural strengthening of Hadwiger's Conjecture arises: is every graph with no dominating $K_t$-model $(t-1)$-colourable? We provide two pieces of evidence for this: (1) It is true for almost every graph, (2) Every graph $G$ with no dominating $K_t$-model has a $(t-1)$-colourable induced subgraph on at least half the vertices, which implies there is an independent set of size at least $\frac{\lvert V(G) \rvert}{2t-2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14299 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dominating $K_t$-Models Illingworth, Freddie Wood, David R. Combinatorics 05C83 A $\textit{dominating $K_t$-model}$ in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leqslant i<j \leqslant t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to $K_t$ being a minor of $G$. We explore in what sense dominating $K_t$-models behave like (non-dominating) $K_t$-models. The two notions are equivalent for $t \leqslant 3$, but are already very different for $t = 4$, since the 1-subdivision of any graph has no dominating $K_4$-model. Nevertheless, we show that every graph with no dominating $K_4$-model is 2-degenerate and 3-colourable. More generally, we prove that every graph with no dominating $K_t$-model is $2^{t-2}$-colourable. Motivated by the connection to chromatic number, we study the maximum average degree of graphs with no dominating $K_t$-model. We give an upper bound of $2^{t-2}$, and show that random graphs provide a lower bound of $(1-o(1))t\log t$, which we conjecture is asymptotically tight. This result is in contrast to the $K_t$-minor-free setting, where the maximum average degree is $Θ(t\sqrt{\log t})$. The natural strengthening of Hadwiger's Conjecture arises: is every graph with no dominating $K_t$-model $(t-1)$-colourable? We provide two pieces of evidence for this: (1) It is true for almost every graph, (2) Every graph $G$ with no dominating $K_t$-model has a $(t-1)$-colourable induced subgraph on at least half the vertices, which implies there is an independent set of size at least $\frac{\lvert V(G) \rvert}{2t-2}$. |
| title | Dominating $K_t$-Models |
| topic | Combinatorics 05C83 |
| url | https://arxiv.org/abs/2405.14299 |