The local vertex anti-magic coloring for certain graph operations
L. Uma, G. Rajasekaran
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.
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}.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · Graph theory and applications