On Coupon Coloring of Cayley Graphs

Abstract

AbstractA k-coupon coloring of a graph G without isolated vertices is an assignment of colors from \([k]=\{1,2,\dots ,k\}\) to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum k for which a k-coupon coloring exists is called the coupon coloring number of G. The Cayley graph \(Cay(\varGamma ,C)\) of a group \(\varGamma \) is a graph with vertex set \(\varGamma \) and edge set \(E(Cay(\varGamma ,C))=\{gh:hg^{-1}\in C\}\), where C is a subset of \(\varGamma \) that is closed under taking inverses and does not contain the identity. Let R be a commutative ring with unity. Then \(Cay(R^+,Z^*(R))\) is denoted by \(\mathbb {CAY}(R)\), where \(R^+\) is the additive group and \(Z^*(R)\) is the non-zero zero-divisors of R. For a natural number n, the generalized Cayley graph, \(\varGamma _{R}^{n}\) is a simple graph with vertex set \(R^n\setminus \{0\}\) and two distinct vertices X and Y are adjacent if and only if there is a lower triangular matrix A over R whose entries on the main diagonal are non-zero and such that \(AX^T=Y^T\) or \(AY^T=X^T\), where \(B^T\) is the transpose of the matrix B. In this paper, we have studied the coupon coloring of \(\mathbb {CAY}({R})\) and generalized Cayley graph \(\varGamma _{R}^n\).KeywordsCoupon coloringCayley graphGeneralized Cayley graph