Distance mutual-visibility coloring: relations with (total) domination, exact distance graphs and graph products

TL;DR

The paper introduces and studies $k$-distance mutual-visibility ($k$DMV) coloring, focusing on the case $k=2$, and connects it to classical graph invariants such as total domination, domination, and exact distance-2 graphs. It establishes fundamental bounds $χ_{μ_k}(G)$ across $k$, shows sharpness for many graph families, and reveals deep links to graph products: lexicographic, strong, and Cartesian products yield both upper and lower bounds that are tight in broad classes. A key finding is the equality $\theta(G^{[\natural 2]})=γ_t(G)$ for isolate-free graphs with girth at least seven, tying $2$DMV coloring to exact distance-2 graphs. The work also provides exact results for several product graphs, characterizations for block graphs, and a set of open problems that prompt further exploration of $k$DMV colorings in complex graph constructions.

Abstract

The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at most $k$ whose internal vertices are not in $S$. In this paper, we combine this with the MV coloring as follows. For any integer $k\geq1$, a $k$DMV coloring of $G$ is a partition of $V(G)$ into $k$DMV sets, and the $k$DMV chromatic number $χ_{μ_k}(G)$ is the minimum cardinality of such a partition. When $k=1$ or $k\ge {\rm diam}(G)$, it equals the clique cover number $θ(G)$ or the MV chromatic number $χ_μ(G)$, respectively. So, our attention is given to $1<k<{\rm diam}(G)$ with $k=2$ producing the most interesting results. We prove that $χ_{μ_2}(G)\le|V(G)|/2$ and present large families of graphs that attain the bound. In addition, $χ_{μ_2}(G)$ is bounded from above by the total domination number $γ_t(G)$ if $G$ is isolate-free, while in graphs $G$ with girth $g(G)\geq7$, $χ_{μ_2}(G)$ is bounded from below by the domination number $γ(G)$. A surprising relation with the exact distance-2 graphs is found, which results in $θ(G^{[\natural2]})=γ_{t}(G)$ for any isolate-free graph $G$ with $g(G)\geq7$. The relation is explored further in lexicographic product graphs, where we prove the sharp inequalities $χ_{μ_{2}}(G\circ H)\leq θ(G^{[\natural2]})\leq θ\big{(}(G\circ H)^{[\natural2]}\big{)}$. We also prove a sharp lower (resp. upper) bound on $χ_{μ_2}$ (resp. $χ_{μ_k}$) for the Cartesian (resp. strong) product of two connected graphs and show that they are widely sharp. Finally, we characterize the block graphs $G$ with $χ_{μ_k}(G)=χ_μ(G)$, where $k={\rm diam}(G)-1$.

Figures (7)

• Figure 1: The graph $G$ with $g(G)=6$, $\gamma(G)=5$ and $\chi_{\mu_2}(G)=3$.
• Figure 2: A tree $T$ with $\gamma_{t}(T)=2\ell+2$ and $\chi_{\mu_{2}}(T)=\ell+2$. Here, $x$ (resp. $y$) is adjacent to $a$ (resp. $b$) leaves and it is an endvertex of $\ell$ copies of the path $P_{4}$.
• Figure 3: A $\chi_{\mu_2}(G)$-coloring of the graph $G$ with $2$ colors.
• Figure 4: The tree $T$, with and $\chi_{\mu_{2}}(T)=3$, mentioned in the proof of Proposition \ref{['pr']}.
• Figure 5: A $3$DMV coloring of $P_{15} \boxtimes K_3$ using $6$ colors. Vertices are partitioned into sets $V_i$, as per the vertices $u_{i}$, each forming a triangle. The subgraphs $G_1$, $G_2$ and $G_3$ are outlined with dashed rounded rectangles.

Theorems & Definitions (33)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof