# The local vertex anti-magic coloring for certain graph operations

## TL;DR

This paper explores local vertex anti-magic colorings for various graph operations and provides partial answers to an open problem in graph theory.

## Contribution

The paper introduces new results on local vertex anti-magic colorings for specific graph operations and products.

## Key findings

- Local vertex anti-magic coloring is proven for even regular circulant bipartite graphs.
- Coloring is established for union of bipartite graphs and join graphs with specific structures.
- Partial answers are given for the open problem regarding the local vertex anti-magic chromatic number of graph products.

## 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); and3.χℓva(P3×H), where H∈{Kr,Km,m}.

χℓva(C2m×C2n);

χℓva(C2m+1×C2n+2); and

χℓva(P3×H), where H∈{Kr,Km,m}.