arXiv subjects

Mirko Hornak

Publications and source records attributed to Mirko Hornak.

3 recordsLinked to original sources

arXiv2021

The achromatic number of the Cartesian product of $K_6$ and $K_q$

Let $G$ be a graph and $C$ a finite set of colours. A vertex colouring $f:V(G)\to C$ is complete if for any pair of distinct colours $c_1,c_2\in C$ one can find an edge $\{v_1,v_2\}\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The achromatic number of $G$ is defined to be the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper $\mathrm{achr}(K_6\square K_q)$ is determined for any integer $q$ such that either $8\le q\le40$ or $q\ge42$ is even.

math.CO↗

arXiv2020

The achromatic number of $K_6\square K_7$ is $18$

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any two distinct colours $c_1,c_2\in C$ there is an edge $\{v_1,v_2\}\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper it is proved that $\mathrm{achr}(K_6\square K_7)=18$. This result finalises the determination of $\mathrm{achr}(K_6\square K_q)$.

math.CO↗

arXiv2020

The achromatic number of $K_6\square K_q$ equals $2q+3$ if $q\ge41$ is odd

Let $G$ be a graph and $C$ a finite set of colours. A vertex colouring $f:V(G)\to C$ is complete provided that for any two distinct colours $c_1,c_2\in C$ there is $v_1v_2\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The achromatic number of $G$ is the maximum number $\mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper it is proved that if $q\ge41$ is an odd integer, then the achromatic number of the Cartesian product of $K_6$ and $K_q$ is $2q+3$.

math.CO↗