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Determining the b-chromatic number of subdivision-vertex neighbourhood coronas

Raúl M. Falcón, M. Venkatachalam, S. Julie Margaret

TL;DR

This work determines the $b$-chromatic number \(\varphi(G)\) for subdivision-vertex neighbourhood (SVN) coronas \(G\boxdot H\) and \(G\boxdot K_n\) when \(G,H\) range over paths, cycles, stars, and complete graphs. It combines tight upper bounds derived from degree-based invariants with explicit, constructive colorings that achieve these bounds, providing complete characterizations (often with precise piecewise formulas) and illustrating them with concrete examples. The results span SVN coronas involving P, C, S, and K families, including many exact values and several notable exceptions, while also addressing the case \(K_n\boxdot G\) under the bound \(m(K_n\boxdot G)\le n+2\). The work lays groundwork for future study of SVN coronas beyond these families and points to open problems when the bound is exceeded, as well as extensions to subdivision-edge neighbourhood coronas. Overall, the paper advances the understanding of how subdivision and vertex-neighborhood operations interact with $b$-colorings in graph products and provides constructive methods useful for both theory and applications.

Abstract

Let $G$ and $H$ be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the $b$-chromatic number of every subdivision-vertex neighbourhood corona $G\boxdot H$ or $G\boxdot K_n$, where $K_n$ is the complete graph of order $n$. It is also established for those graphs $K_n\boxdot G$ having $m$-degree not greater than $n+2$. All the proofs are accompanied by illustrative examples.

Determining the b-chromatic number of subdivision-vertex neighbourhood coronas

TL;DR

This work determines the -chromatic number \(\varphi(G)\) for subdivision-vertex neighbourhood (SVN) coronas and when range over paths, cycles, stars, and complete graphs. It combines tight upper bounds derived from degree-based invariants with explicit, constructive colorings that achieve these bounds, providing complete characterizations (often with precise piecewise formulas) and illustrating them with concrete examples. The results span SVN coronas involving P, C, S, and K families, including many exact values and several notable exceptions, while also addressing the case under the bound \(m(K_n\boxdot G)\le n+2\). The work lays groundwork for future study of SVN coronas beyond these families and points to open problems when the bound is exceeded, as well as extensions to subdivision-edge neighbourhood coronas. Overall, the paper advances the understanding of how subdivision and vertex-neighborhood operations interact with -colorings in graph products and provides constructive methods useful for both theory and applications.

Abstract

Let and be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the -chromatic number of every subdivision-vertex neighbourhood corona or , where is the complete graph of order . It is also established for those graphs having -degree not greater than . All the proofs are accompanied by illustrative examples.
Paper Structure (7 sections, 24 theorems, 42 equations, 24 figures)

This paper contains 7 sections, 24 theorems, 42 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SVN corona of paths
  4. SVN corona of cycles
  5. SVN corona of stars
  6. SVN corona of complete graphs
  7. Conclusion and further work

Key Result

Lemma 1

Irving1999 Let $G$ be a graph. Then, $\chi(G) \leq \varphi(G) \leq \Delta (G) + 1$.

Figures (24)

  • Figure 1: SVN corona product.
  • Figure 2: Optimal $b$-chromatic coloring of $P_{10}\boxdot P_3$.
  • Figure 3: Optimal $b$-chromatic coloring of $P_4\boxdot P_4$.
  • Figure 4: Optimal $b$-chromatic colorings of $P_{7}\boxdot P_4$ and $P_{8}\boxdot P_4$.
  • Figure 5: Optimal $b$-chromatic coloring of $P_3\boxdot C_4$.
  • ...and 19 more figures

Theorems & Definitions (45)

  • Lemma 1
  • Proposition 2
  • Lemma 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 35 more