Orthogonal Labeling for Some Different Infinite Graph Classes

Abstract

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. Graph labeling is one of the fascinating areas of graph theory with wide ranging applications. A collection $\mathcal{A}$ of isomorphic copies of a given subgraph $G$ of $T$ is said to be an orthogonal double cover (ODC) of the graph $T$ by $G$, if every edge of $T$ belongs to exactly two members of $\mathcal{A}$ and any two different elements from $\mathcal{A}$ share at most one edge. The existence of an orthogonal labeling of a graph $G$ with respect to a certain group implies the existence of the cyclic orthogonal double cover of the circulant graphs on that group. In this paper, we prove the existence of orthogonal labeling for some different infinite graph classes and hence, the existence of the cyclic orthogonal double cover of some different infinite circulant graphs.