AVD Total Coloring of Central Graphs, Subdivision Graphs, and the Join of Graphs

Abstract: In 2020, Panda, Verma, and Keerti asked whether the central graph of every graph satisfies the AVD-total coloring conjecture. In this paper, we verify the conjecture for central graphs of regular graphs, complete bipartite graphs, graphs that can be expressed as the join of two graphs of the same order, and several other graph classes, thereby providing partial progress towards this open problem. We further determine the AVD-total chromatic number of subdivision graphs and establish new results on the AVD-total coloring of joins of graphs.

• The paper determines the AVD-total chromatic numbers of subdivision, central, and join graphs, clearly classifying them as AVD-Type 1 or 2 based on structural properties.
• The authors employ combinatorial methods and anti-circulant Latin squares to construct colorings that ensure distinct adjacent vertex color sets.
• The results have practical implications for network labeling and offer new strategies that advance the resolution of the AVD-total coloring conjecture.

AVD Total Coloring of Central Graphs, Subdivision Graphs, and the Join of Graphs

Introduction and Background

The study of graph colorings has been a central topic in graph theory, with significant focus on chromatic numbers for both vertices and edges. The concept of adjacent vertex distinguishing total coloring (AVD-total coloring) extends the traditional notion of proper total coloring to impose an additional adjacency constraint: for every edge $uv$ in a graph $G$, the color sets at $u$ and $v$—including their color and the colors of their incident edges—must be distinct. The minimal number of colors required to achieve an AVD-total coloring is denoted $\chi''_{a}(G)$, the AVD-total chromatic number.

A major conjecture in this area, the AVD-total coloring conjecture (AVD-TCC), asserts that $\chi''_{a}(G)\leq \Delta(G)+3$ for any graph $G$, where $\Delta(G)$ is the maximum degree of $G$. While prior works have established the conjecture for several prominent graph classes—including cubic graphs, $\Delta=4$ graphs, hypercubes, and split graphs—its validity for broader graph operations and derived classes remains open.

This paper conducts a comprehensive investigation into the AVD-total chromatic numbers of three important families of graphs derived from a base graph $G$0: the subdivision graph $G$1, the central graph $G$2, and joins of graphs. Structural results for each family are established, and the AVD-TCC is resolved or partially resolved for broad classes therein.

Subdivision and Central Graphs: Definitions and Results

The subdivision graph $G$3 is obtained by inserting a vertex into each edge of $G$4. The central graph $G$5 is constructed by starting from $G$6 and then adding all the edges from the complement $G$7 of $G$8, thereby connecting all nonadjacent vertex pairs of $G$9.

The paper first determines $u$0 exactly for all connected $u$1:

• $u$2 is AVD-Type~1 (i.e., $u$3) if and only if $u$4 or $u$5.
• Otherwise (i.e., for paths and cycles of small order), $u$6 is AVD-Type~2 ($u$7).

For proof, the authors exploit existing results on AVD-total coloring of bipartite graphs—citing the fact that $u$8 is always bipartite—and classic results for $u$9 and $v$0. The construction explicitly uses proper edge coloring and careful assignment of new colors to subdivision vertices so the color sets at adjacent vertices always differ.

Central graphs are more complex as the addition of $v$1 edges tends to make them highly connected. Focusing on regular graphs $v$2, the following affirmative results are obtained:

• If the order $v$3 of $v$4 is even, $v$5 is always AVD-Type~2, so $v$6.
• If $v$7 is odd, $v$8 satisfies the AVD-TCC, i.e., $v$9.

The constructions rely on anti-circulant idempotent commutative Latin squares, ensuring the required coloring distinctions and leveraging combinatorial structure inherent to regular graphs for efficient colorings.

Notably, for central graphs of complete bipartite graphs, the AVD-TCC also holds, generalizing prior results.

AVD-Total Coloring of Joins and Related Constructions

The paper makes substantial progress on the AVD-total coloring of join graphs (denoted $\chi''_{a}(G)$0). The key claims are as follows:

• For graphs $\chi''_{a}(G)$1 and $\chi''_{a}(G)$2 with $\chi''_{a}(G)$3, $\chi''_{a}(G)$4 and $\chi''_{a}(G)$5, if $\chi''_{a}(G)$6 is odd, $\chi''_{a}(G)$7 satisfies the AVD-TCC.
• When $\chi''_{a}(G)$8 and $\chi''_{a}(G)$9 have equal order $\chi''_{a}(G)\leq \Delta(G)+3$0, and $\chi''_{a}(G)\leq \Delta(G)+3$1 is Type~1, $\chi''_{a}(G)\leq \Delta(G)+3$2 satisfies the AVD-TCC.

These results are achieved by constructing colorings that use anti-circulant Latin squares of appropriate orders, allowing construction of colorings with the necessary distinguishing properties. The argument confirms that color sets at endpoints of every edge are distinct and the coloring uses no more than $\chi''_{a}(G)\leq \Delta(G)+3$3 colors.

The extension to central graphs of joins and specific families such as complete bipartite, path, and cycle joins is handled analogously, and the AVD-TCC is established in all these cases.

Theoretical and Practical Implications

The paper not only resolves several previously open cases of the AVD-TCC but also introduces robust combinatorial tools for coloring regular, bipartite, and join-derived families. The explicit use of commutative Latin squares as color assignment matrices represents a methodologically significant step that may aid future colorings in more general settings.

Practically, AVD-total colorings correspond to labeling scenarios where adjacent network nodes (and their incident connections) require unique signatures or frequencies, such as in communication protocols, resource assignment, or network identification schemes, especially in networks built from composition or subdivision of smaller graphs.

Theoretically, these results bring the community closer to a full resolution of the AVD-TCC for all central and join-structured graphs. The classification result for subdivision graphs provides a clean dichotomy depending on the degree of the original graph, which informs the construction of more complex graph families whose AVD-total chromatic number can be controlled or tightly bounded.

Open Questions and Future Directions

Significant progress notwithstanding, two concrete open directions remain:

• Whether the AVD-TCC holds for all central graphs $\chi''_{a}(G)\leq \Delta(G)+3$4, regardless of $\chi''_{a}(G)\leq \Delta(G)+3$5.
• Classification of non-regular graphs $\chi''_{a}(G)\leq \Delta(G)+3$6 for which $\chi''_{a}(G)\leq \Delta(G)+3$7 is AVD-Type~1.

Future work could leverage further algebraic and combinatorial techniques (perhaps extensions of the Latin square paradigm) to approach these problems in full generality, contribute new coloring algorithms, and examine computational complexity of determining or approximating $\chi''_{a}(G)\leq \Delta(G)+3$8 on these and related families.

Conclusion

The paper rigorously determines the AVD-total chromatic number for subdivision graphs and establishes the AVD-TCC for wide classes of central graphs and joins of graphs, utilizing combinatorial constructions based on Latin squares. The structural theorems and proof techniques provide both precise classifications and practical coloring methodologies for graph classes of contemporary and foundational interest in combinatorial mathematics and its applications.

For researchers interested in further developing or applying total coloring results, these advances substantially widen the toolkit for tackling complex, highly composite graph classes and for approaching deep unsolved conjectures in the coloring theory landscape.