Redicoloring some classes of circulant tournaments

TL;DR

This work addresses the connectivity of dicoloring graphs for digraphs, showing that no single function of the dichromatic number $\\vec{\\chi}(D)$ can universally guarantee connectivity of $\\mathcal{D}_k(D)$ for all $D$ and $k\\geq\phi(\\vec{\\chi})$. Focusing on two infinite families of circulant tournaments, the authors establish connectivity criteria and explicit diameter bounds for $\\mathcal{D}_k$ in these families, and they verify a concrete case: the Paley-like tournament $ST_7$ (i.e., $\\vec{C}_{7}(1,2,4)$) yields a connected $\\mathcal{D}_k$ with diameter $8$ for all $k\\geq3$. The paper provides precise results for the cyclic case $\\vec{C}_{2n+1}\\langle \emptyset\\rangle$ and the non-symmetric case $\\vec{C}_{2n+1}\\langle n\\rangle$, including conditions under which $\\mathcal{D}_k$ is disconnected and the sizes of the isolated components. Overall, these findings advance the understanding of dicoloring reconfiguration in digraphs and offer tight bounds crucial for algorithmic recolorings in circulant structures.

Abstract

Given a digraph $D$ with no loops, the \textit{dicoloring graph} of $D$, denoted by $\mathcal{D}_k(D)$, is the graph whose vertices are the acyclic $k$-colorings of $D$ and two colorings are adjacent in $\mathcal{D}_k(D)$ if they differ in color on exactly one vertex. In this paper, we prove that there is no expression $φ(\vecχ)$ in terms of the dichromatic number $\vecχ$, such that the graph $\mathcal{D}_k(D)$ is connected for all graphs $D$ and integers $k\geq φ(\vecχ)$. We give conditions for the dicoloring graph of two infinite families of circulant tournaments to be connected, and we provide upper bounds for its diameter. In particular, for the Payley tournament $\vec{C}_{7}(1,2,4)$, also known as $ST_7$, we prove that $\mathcal{D}_k(\vec{C}_{7}(1,2,4))$ is connected and has diameter 8, for each $k\geq 3$.

Figures (3)

• Figure 1: A 2-dichromatic digraph $D$, with a disconnected 2-dicoloring graph.
• Figure 2: Case 1. $|C_1^{\alpha}| \geq 3$ whenever $k=3$ and $\alpha(a+1)=\alpha(a+n)$. Color 1 is green, color $l$ is red and color $h$ is blue. The vertices for which we do not know the color assigned to them are represented in black.
• Figure 3: Case 1. $|C_1^{\alpha}| \geq 3$ whenever $k=3$ and $\alpha(a+1)\neq \alpha(a+n)$. Color 1 is green, color $l$ is red and color $h$ is blue. The vertices for which we do not know the color assigned to them are represented in black.

Theorems & Definitions (30)

• Proposition 1
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• proof
• Corollary 5
• Theorem 6: Neumann-Lara and Urrutia NEUMANNLARA198483
• Proposition 7
• proof