(Injective) hom-complexity between graphs

Abstract

We present the notion of hom-complexity, $\text{C}(G;H)$, for two graphs $G$ and $H$, along with basic results for this numerical invariant. This invariant $\text{C}(G;H)$ is a number that measures the \aspas{complexity} of the question: when is there a homomorphism $G\to H$? More precisely, $\text{C}(G;H)$ is the least positive integer $k$ such that there are $k$ different subgraphs $G_j$ of $G$ such that $G=G_1\cup\cdots\cup G_k$, and for each $G_j$, there is a homomorphism $G_j\to H$. Likewise, we introduce the notion of injective hom-complexity, $\text{IC}(G;H)$. The (injective) hom-complexity is a graph invariant. Additionally, these invariants can be used to show the nonexistence of homomorphisms. We explore the sub-additivity of (injective) hom-complexity and study products. We describe bounds for the hom-complexity in terms of chromatic number $\chi$ and clique number $\omega$. We provide the formula \[\text{C}(G;H)=\lceil\log_{\chi(H)}\chi(G)\rceil\] whenever $\omega(H)=\chi(H)$. For example, we obtain $\text{C}(G;K_\ell)=\lceil\log_{\ell}\chi(G)\rceil$. Moreover, we discuss a connection between the (injective) hom-complexity and several well-known covering numbers. For instance, we provide a lower bound for the clique covering number in terms of the injective hom-complexity. Additionally, we show that the hom-complexity $\mathrm{C}(G;K_{\ell})$ coincides with the $\ell$-particity $\beta_\ell(G)$ of $G$, and the hom-complexity $\mathrm{C}(K_n;K_{2})$ coincides with the bipartite dimension $\mathrm{d}(K_n)$ of $K_n$. As a consequence, we recover the well-known formulas $\beta_\ell(G)=\lceil\log_{\ell}\chi(G)\rceil$ and $\mathrm{d}(K_n)=\lceil\log_{2}n\rceil$.