%0 Journal Article
%T A note on δ^(k)-colouring of the Cartesian product of some graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Naduvath, Sudev
%A Ellumkalayil, Merlin Thomas
%D 2022
%\ 06/01/2022
%V 7
%N 1
%P 113-120
%! A note on δ^(k)-colouring of the Cartesian product of some graphs
%K Improper colouring
%K near proper colouring
%K δ^(k)-colouring
%K bad edge
%R 10.22049/cco.2021.27114.1211
%X The chromatic number, $chi(G)$ of a graph $G$ is the minimum number of colours used in a proper colouring of $G$. In an improper colouring, an edge $uv$ is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph $G$, then colouring the graph with the available colours lead to bad edges in $G$. The number of bad edges resulting from a $delta^{(k)}$-colouring of $G$ is denoted by $b_{k}(G)$. In this paper, we use the concept of $delta^{(k)}$-colouring and determine the number of bad edges in Cartesian product of some graphs.
%U http://comb-opt.azaruniv.ac.ir/article_14225_7671b9be902fe5288eaea7c2a4aa2762.pdf