Independent mutual-visibility coloring and related concepts

TL;DR

The paper introduces and analyzes independent mutual-visibility colorings in graphs, formalizing the IMV number $\mu_i(G)$ and the IMV chromatic number $\chi_{\mu_i}(G)$ alongside their non-independent counterparts $\mu(G)$ and $\chi_{\mu}(G)$. It establishes deep links to Ramsey theory via subdivision graphs, proves NP-completeness for computing these parameters, and provides tight bounds and exact values for key graph families, including trees, cycles, and lexicographic, Cartesian, and strong products. Notably, it proves $\chi_{\mu_i}(S(K_n))$ and $\chi_{\mu}(S(K_n))$ are tied to partitions avoiding $K_4$ subgraphs, yielding Ramsey-number-based bounds, and shows $\chi_{\mu_i}(G\circ H)=\chi(G\circ H)$ with $\mu_i(G\circ H)=\alpha(G)\alpha(H)$ under broad conditions. The work also delivers exact results for several product graphs (e.g., $P_t\boxtimes P_r$, trees under Cartesian/strong products) and advances our understanding of MV/IMV colorings’ structural and computational complexity, while highlighting numerous open questions about subdivision graphs and broader graph classes.

Abstract

Given a graph $G$, a subset $M\subseteq V(G)$ is a mutual-visibility (MV) set if for every $u,v\in M$, there exists a $u,v$-geodesic whose internal vertices are not in $M$. We investigate proper vertex colorings of graphs whose color classes are mutual-visibility sets. The main concepts that arise in this investigation are independent mutual-visibility (IMV) sets and vertex partitions into these sets (IMV colorings). The IMV number $μ_{i}$ and the IMV chromatic number $χ_{μ_{i}}$ are defined as maximum and minimum cardinality taken over all IMV sets and IMV colorings, respectively. Along the way, we also continue with the study of MV chromatic number $χ_μ$ (as the smallest number of sets in a vertex partition into MV sets), which was initiated in an earlier paper. We establish a close connection between the (I)MV chromatic numbers of subdivisions of complete graphs and Ramsey numbers $R(4^k;2)$. From the computational point of view, we prove that the problems of computing $χ_{μ_{i}}$ and $μ_{i}$ are NP-complete, and that it is NP-hard to decide whether a graph $G$ satisfies $\imv(G)=α(G)$ where $α(G)$ is the independence number of $G$. Several tight bounds on $χ_{μ_{i}}$, $χ_μ$ and $μ_{i}$ are given. Exact values/formulas for these parameters in some classical families of graphs are proved. In particular, we prove that $χ_{μ_{i}}(T)=χ_μ(T)$ holds for any tree $T$ of order at least $3$, and determine their exact formulas in the case of lexicographic product graphs. Finally, we give tight bounds on the (I)MV chromatic numbers for the Cartesian and strong product graphs, which lead to exact values in some important families of product graphs.

Figures (2)

• Figure 1: The graph $G^*$, which is the complete graph $K_4$ with subdivided edges. An optimal IMV $3$-coloring is drawn.
• Figure 2: An illustration of the graph $G$, constructed in the proof of Theorem \ref{['Hard']}, with $a=b=4$. Here, $U=\{u_{1},u_{2},u_{3},u_{4}\}$, $C_{1}=\{u_{1},u_{2},u_{3}'\}$, $C_{2}=\{u_{1}',u_{2}',u_{4}\}$, $C_{3}=\{u_{2}',u_{3},u_{4}\}$ and $C_{4}=\{u_{1}',u_{3}',u_{4}'\}$. Notice that ${(}f(u_{1}),f(u_{2}),f(u_{3}),f(u_{4}){)}={(}True,False,True,False{)}$ is a satisfying truth assignment for $C$ and $\bigcup_{i=1}^{4}\{c_{i}',r_{i},s_{i}\}\cup \{z\}\cup \{p_{1}',p_{2},p_{3}',p_{4}\}$ is an IMV set in $G$ of cardinality $\alpha(G)=3a+b+1=17$.

Theorems & Definitions (50)

• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 4
• Theorem 5
• proof
• Theorem 6
• proof
• Proposition 7