The distinguishing number and the distinguishing index of co-normal product of two graphs

arXiv1707.06533MaRDI QIDQ6289244

Samaneh Soltani, Saeid Alikhani

Publication date: 19 July 2017

Abstract: The distinguishing number (index) $D(G)$ ($D^{\prime }(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The co-normal product $GstarH$ of two graphs $G$ and $H$ is the graph with vertex set $V(G)imesV(H)$ and edge set $(x_1,x_2),(y_1,y_2)|x_1{y}_{1}inE(G)mor{x}_2{y}_{2}inE(H)$. In this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. We prove that for every $kgeq3$, the $k$-th co-normal power of a connected graph $G$ with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two.

Mathematics Subject Classification ID

Coloring of graphs and hypergraphs (05C15) Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) (05C60)

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