Dominating K t -Models

A dominating $K_t$-model in a graph $G$ is a sequence $\left(T_1,\dots ,T_t\right)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1⩽i<j⩽t$ 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 $t⩽3$ 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 $(1-o\left(1\right))t\log t$, which we conjecture is asymptotically tight. This result is in contrast to the $K_t$-minor-free setting, where the maximum average degree is $\Theta \left(t\sqrt{\log t}\right)$. A natural strengthening of Hadwiger's conjecture arises: Is every graph with no dominating $K_t$-model $\left(t-1\right)$-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 $\left(t-1\right)$-colourable induced subgraph on at least half the vertices, which implies there is an independent set of size at least $\frac{\left|V\left(G\right)\right|}{2t-2}$.