Vertex equitable labeling for ladder and snake related graphs

Abstract

Let GG be a graph with p vertices and q edges and A=0,1,2,⋅⋅⋅,q2A=0,1,2,···,q2. A vertex labeling f:V(G)→Af:V(G)→A induces an edge labeling f∗f∗ defined by f∗(uv)=f(u)+f(v)f∗(uv)=f(u)+f(v) for all edges uvuv. For a∈Aa∈A, let vf(a)vf(a) be the number of vertices vv withf(v)=af(v)=a. A graph GG is said to be vertex equitable if there exists a vertex labeling f such that for all aa and bb in A, |vf(a)−vf(b)|≤1|vf(a)−vf(b)|≤1 and the induced edge labels are 1,2,3,⋅⋅⋅,q1,2,3,···,q. In this paper, we prove that triangular ladder TLn,Ln⊙mK1,Qn⊙K1,TLn⊙K1TLn,Ln⊙mK1,Qn⊙K1,TLn⊙K1 and alternate triangular snake A(Tn)A(Tn) are vertex equitable graphs.