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Determining the Locating Rainbow Connection Number of Vertex-Transitive Graphs

Ariestha Widyastuty Bustan, ANM Salman, Pritta Etriana Putri

Abstract

The locating rainbow connection number of a graph is defined as the minimum number of colors required to color vertices such that every two vertices there exists a rainbow vertex path and every vertex has a distinct rainbow code. This rainbow code signifies a distance between vertices within a given set of colors in a graph. This paper aims to determine the locating rainbow connection number for vertex-transitive graphs. Three main theorems are derived, focusing on the locating rainbow connection number for some vertex-transitive graphs

Determining the Locating Rainbow Connection Number of Vertex-Transitive Graphs

Abstract

The locating rainbow connection number of a graph is defined as the minimum number of colors required to color vertices such that every two vertices there exists a rainbow vertex path and every vertex has a distinct rainbow code. This rainbow code signifies a distance between vertices within a given set of colors in a graph. This paper aims to determine the locating rainbow connection number for vertex-transitive graphs. Three main theorems are derived, focusing on the locating rainbow connection number for some vertex-transitive graphs
Paper Structure (6 sections, 9 theorems, 1 equation, 8 figures)

This paper contains 6 sections, 9 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Locating Rainbow Connection Number of $2$-Regular Graphs
  4. The Locating Rainbow Connection Number of $(n,t)$-Regular Graphs for $t=\{2,3\}$
  5. The Locating Rainbow Connection Number of $(n,2)$-Regular Graphs
  6. The Locating Rainbow Connection Number of $(n,3)$-Regular Graphs

Key Result

Lemma 1

bustan2021locating Let $c$ be a locating rainbow coloring of $G$. Let $u$ and $v$ be two distinct vertices of $G$. If $d(u,x)=d(v,x)$ for all $x \in V(G)-\{u,v\}$, then $c(u)\neq c(v)$.

Figures (8)

  • Figure 1: Locating rainbow $3$-colorings of (a) $C_3$, (b) $C_4$, (c) $C_5$, (d) $C_6$, and (e) $C_7$.
  • Figure 2: Rainbow codes of (a) $C_8$, (b) $C_9$, and (c) $C_{10}$.
  • Figure 3: Rainbow codes of (a) $C_{11}$, (b) $C_{12}$, (c) $C_{13}$, and $C_{15}$.
  • Figure 4: (a) $R_{(12,1)}$, (b) $R_{(12,2)}$
  • Figure 5: Rainbow codes of $R_{(16,2)}$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 3 more