arXiv subjects

Mirko Horňák

Publications and source records attributed to Mirko Horňák.

2 recordsLinked to original sources

arXiv2022

3-facial edge-coloring of plane graphs

An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most $3\ell + 1$ colors suffice for an $\ell$-facial edge-coloring of any plane graph. The conjecture has only been confirmed for $\ell \le 2$, and in this paper, we prove its validity for $\ell = 3$.

math.CO↗

arXiv2022

On the achromatic number of the Cartesian product of two complete graphs

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any $c_1,c_2\in C$ with $c_1\ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. Let $G_1\square G_2$ denote the Cartesian product of graphs $G_1$ and $G_2$. In the paper $\mathrm{achr}(K_{r^2+r+1}\square K_q)$ is determined for an infinite number of $q$s provided that $r$ is a finite projective plane order.

math.CO↗