Homomorphisms of planar $(m, n)$-colored-mixed graphs to planar targets

TL;DR

This paper investigates the existence and construction of universal planar graphs for $(m,n)$-colored-mixed graphs with certain girth constraints, revealing non-existence in many cases and providing minimal examples otherwise.

Contribution

It proves the non-existence of planar $P_g^{(m, n)}$-universal graphs for $2m+n \\geq 3$ and constructs minimal such graphs for other parameter ranges.

Findings

Universal planar graphs do not exist for $2m+n \\geq 3$.

Minimal planar $P_g^{(m, n)}$-universal graphs are identified for cases where they exist.

The results delineate the boundary between existence and non-existence of such universal graphs.

Abstract

An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a morphism $\varphi:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m, n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m, n)$-colored-mixed graphs with girth at least $g$) admits a homomorphism to $T$. We show that planar $P_g^{(m, n)}$-universal graphs do not exist for $2m+n\ge3$ (and any value of $g$) and find a minimal (in the number vertices) planar $P_g^{(m, n)}$-universal graphs in the other cases.