Some new results on colour-induced signed graphs

Abstract

A signed graph is a graph in which positive or negative signs are assigned to its edges. We consider equitable colouring and Hamiltonian colouring to obtain induced signed graphs. An equitable colour-induced signed graph is a signed graph constructed from a given graph in which each edge uv receives a sign (−1)|c(v)−c(u)|,where c is an equitable colouring of vertex v. A Hamiltonian colour-induced signed graph is a signed graph obtained from a graph G in which for each edge e = uv, the signature function σ(uv)=(−1)|c(v)−c(u)|, gives a sign such that, |c(u)− c(v)| ≥ n − 1 − D(u, v) where c is a function that assigns a colour to each vertex satisfying the given condition. This paper discusses the properties and characteristics of signed graphs induced by the equitable and Hamiltonian colouring of graphs.