Some Difference Graphs

TL;DR

This paper investigates difference labelings for various standard graphs, proving certain families are difference graphs and introducing new labelings, with implications for classifying specific graphs like complete bipartite graphs.

Contribution

It establishes difference labelings for several graph families and introduces a corollary that aids in classifying graphs as difference graphs or not.

Findings

Proved that Star, Butterfly, Bistar, umbrella, and Olive tree are difference graphs.

Introduced difference labelings for specific snake graphs.

Provided a method to determine non-difference graphs like K_{2,4}.

Abstract

In this paper, we discuss difference labeling of some standard families of graphs. We prove that Star, Butterfly, Bistar, umbrella and Olive tree are difference graphs. We also introduce difference labelings for some snakes (double triangular snake, irregular triangular snake, alternate $C_n$ snake). Furthermore we introduce a corollary helps us to find a unique difference labeling for the complete graph $K_3$ and all forms of difference labeling for the Star graph. Also this corollary can be used to prove that the complete bipartite graph $K_{2,4}$ is not a difference graph but the proof is very lengthy.