Coloring the square of the Cartesian product of two cycles

Abstract

International audienceThe square $G^2$ of a graph $G$ is defined on the vertex set of $G$ in such a way that distinct vertices with distance at most two in $G$ are joined by an edge. We study the chromatic number of the square of the Cartesian product $C_m\Box C_n$ of two cycles and show that the value of this parameter is at most 7 except when $m=n=3$, in which case the value is 9, and when $m=n=4$ or $m=3$ and $n=5$, in which case the value is 8. Moreover, we conjecture that whenever $G=C_m\Box C_n$, the chromatic number of $G^2$ equals $\lceil mn/\alpha(G^2) \rceil$, where $\alpha(G^2)$ denotes the size of a maximal independent set in $G^2$

The incidence chromatic number of toroidal grids

“…We first consider the case when m ≡ 0 (mod 3). We have proved in [14] the following: Proposition 10 If k ≥ 1, n ≥ 3 and n even, then χ(T 2 3k,n ) ≤ 6. Here we prove:…”

“…The main idea is to use a pattern for coloring the square of a toroidal grid in order to get an incidence coloring of this toroidal grid. However, as shown in [14], the squares of toroidal grids are not all 6-colorable. Therefore, we shall use the notion of a quasi-pattern which corresponds to a vertex 6-coloring of the square of a subgraph of a toroidal grid obtained by deleting some edges (namely those edges that cause a conflict when transforming a vertex coloring to its corresponding incidence coloring).…”

Coloring the Dth Power of The Cartesian Product of Two Cycles and Two Paths

“…Chiang and Yan studied the chromatic number of the square of Cartesian products of paths and cycles [1]. Also Sopena and Wu studied the chromatic number of the square of Cartesian product C m C n of two cycles and showed that this value is at most 7 except when (m, n) is (3,3), in such case the value is 9 and when (m, n) is (4,4) or (3,5), the chromatic number is 8 [8]. In [7], Selvakumar and Nithya represented the chromatic number of some graphs, so that in this coloring no two vertices have distance two get the same color.…”

L(1,1)-Labeling of Direct Product of any Path and Cycle

“…The L(h, k)−labeling of the Cartesian product G2H has been extensively investigated with λ k h (G2H) obtained for various types of graphs G and H, while numerous upper and lower bounds have been suggested (see [8] [22]). Most of the work on L(h, k) labeling consider h = 2 and k = 1; although Chiang and Yan in [7] and Georges and Mauro in [10] worked on the L(1, 1)labeling of Cartesian products of paths and cycles and Sopena and Wu in [20] worked on Cartesian products of cycles. In case of direct product graphs, Jha et al [15], established λ 1 2 (C m × C n ) for some values of m and n.…”