The local vertex anti-magic coloring for certain graph operations

Abstract

This work proves the local vertex anti-magic coloring of even regular circulant bipartite graphs C(m;L). Let G be either Kr,r or Kr,r−F, F is a 1-factor. Also, we discover the local vertex anti-magic coloring for union of bipartite graphs; join graphs G∨H, where H∈{Or,Kr,Cr,Kr,s}; and the upper bound of corona product G⊙Or. It was a problem Lau and Shiu (2023) [1] that: For any G1 and G2, determine χℓva(G1×G2). We give partial answer to this problem by proving the followings: 1. χℓva(C2m×C2n); 2. χℓva(C2m+1×C2n+2); and 3. χℓva(P3×H), where H∈{Kr,Km,m}.

Keywords

• Local anti-magic coloring
• Circulant graphs
• Complete bipartite graphs
• Null graphs
• Join graphs
• Complete graphs