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

Area of Science:

• Graph Theory
• Combinatorics
• Discrete Mathematics

Background:

• Local vertex anti-magic coloring is a graph labeling technique where adjacent vertices must have distinct sums of incident edge labels.
• Even regular circulant bipartite graphs and graph operations like unions, joins, and corona products are significant structures in graph theory.

Purpose of the Study:

• To determine the local vertex anti-magic coloring of even regular circulant bipartite graphs.
• To investigate the local vertex anti-magic coloring for graph unions, join graphs, and establish bounds for corona products.
• To provide partial solutions to an open problem concerning local vertex anti-magic coloring for specific graph classes.

Main Methods:

• The study employs constructive proof techniques to establish the existence and properties of local vertex anti-magic colorings.
• Analysis involves examining graph structures, including circulant graphs, bipartite graphs, and their compositions (unions, joins, corona products).
• Specific graph parameters and properties are utilized to derive coloring conditions and bounds.

Main Results:

• The local vertex anti-magic coloring is proven for even regular circulant bipartite graphs $C\_\{2n\}(2,\; 4,\; ...,\; 2n-2)$ and $C\_\{2n\}(1,\; 3,\; ...,\; 2n-1)$ with a 1-factor.
• Local vertex anti-magic coloring is established for the union of bipartite graphs, join graphs $G\; abla\; H$ where $H$ is a complete graph $K\_n$, and bounds for the corona product are determined.
• Partial answers are provided to the problem of determining $Z(G)$ for $G\; =\; C\_n(S)$ and $G\; =\; C\_n[K\_m]$.

Conclusions:

• The research successfully characterizes local vertex anti-magic coloring for several classes of graphs, expanding known results.
• The findings contribute to understanding graph labeling problems and offer partial solutions to open questions in the field.
• This work deepens the theoretical understanding of vertex anti-magic coloring in complex graph structures.