Mutual-Visibility Chromatic Number

• Mutual-Visibility Chromatic Number is defined as the minimum number of colors needed to partition a graph's vertices into sets where each pair is connected by a shortest path whose internal vertices avoid the color.
• It generalizes classical vertex coloring by replacing independence with geodesic visibility, leading to novel bounds and NP-completeness results even for graphs with small diameters.
• Exact formulas and extremal bounds have been derived for specific graph families, and the invariant is closely linked to graph products and Ramsey-theoretic concepts.

The mutual-visibility chromatic number, denoted $\chi_\mu(G)$, is an invariant of finite graphs that quantifies the minimum number of colors required to partition the vertex set into classes in which every pair of vertices is connected by a shortest path whose internal vertices are not assigned that color. This property generalizes classical vertex coloring by replacing independence with geodesic visibility within each color class. The recent foundational works (Babu et al., 13 Dec 2025, Brešar et al., 7 May 2025), and (Klavžar et al., 2024) have established its formal definition, complexity, extremal bounds, exact formulas for families of graphs, and links to other combinatorial parameters.

1. Precise Definition and Core Properties

Given a connected graph $G=(V,E)$, a subset $S \subseteq V$ is a mutual-visibility set if for any pair $x, y \in S$, there exists a shortest $x$–$y$ path whose internal vertices all lie outside $S$. Formally, a coloring $c:V \to [k]$ is a mutual-visibility coloring if for each color $i$, the set $S_i = c^{-1}(i)$ satisfies: for all $x, y \in S_i$, there exists a shortest $x$–$y$ path $P$ in $G$ such that $\operatorname{Int}(P) \cap S_i = \emptyset$ (Babu et al., 13 Dec 2025). The mutual-visibility chromatic number is thus

$$\chi_\mu(G) = \min\{ k \mid V(G) \text{ admits a partition into } k \text{ mutual-visibility sets} \}.$$

For geodetic graphs (graphs with unique shortest paths), at most two vertices per color are possible on a diameter path, yielding $\chi_\mu(G) \ge \lceil(\mathrm{diam}(G)+1)/2\rceil$ (Klavžar et al., 2024).

2. Computational Complexity and NP-Completeness

The decision problem for mutual-visibility coloring is to determine, for given $G$ and $k$, whether $\chi_\mu(G) \leq k$. This problem is NP-complete for $k=2$ even when restricted to graphs of diameter four (Babu et al., 13 Dec 2025). The reduction from NAE-3SAT utilizes gadgets (notably the $H_n$ construction) that enforce two mutually visible color classes whose internal structure mirrors the logical constraints of NAE-3SAT. This shows the nontriviality of the constraint and the impossibility of efficient algorithms for general input unless P = NP.

The related independent mutual-visibility chromatic number $\chi_{\mu_i}(G)$—where each color class is both independent and mutual-visible—is also NP-complete to compute (Brešar et al., 7 May 2025). In addition, deciding equality $\imv(G) = \alpha(G)$ is NP-hard, establishing the intrinsic complexity even in restricted graph families.

3. Exact Formulas and Extremal Bounds

Numerous sharp bounds and explicit formulas have been established:

Graph Family	$\chi_\mu(G)$	Reference
Path $P_n$	$\lceil(n+1)/2\rceil$	(Klavžar et al., 2024, Brešar et al., 7 May 2025)
Cycle $C_n$ ($n\ge 6$)	$\lfloor n/3 \rfloor$	(Brešar et al., 7 May 2025)
Tree $T$	$\operatorname{rad}(T)+1$ if even diameter; $\operatorname{rad}(T)$ if odd	(Klavžar et al., 2024)
Glued binary trees $GT(r)$	Piecewise: $2(r-i)+3$ or $2(r-i)+2$ for integer $i$ depending on $r$	(Babu et al., 13 Dec 2025)
Block graph, $\text{diam}=d$	$\lceil(d+1)/2\rceil$	(Klavžar et al., 2024, Babu et al., 11 Oct 2025)

The extremal lower bound $\chi_\mu(G) \ge \lceil |V(G)|/\mu(G)\rceil$ holds for general graphs ($\mu(G)$ is the maximal mutual-visibility set) (Klavžar et al., 2024).

For Cartesian products of cliques, $\chi_\mu(K_n \square K_n) = \Theta(\sqrt{n})$ (Klavžar et al., 2024). For certain highly symmetric constructions—e.g., glued $t$-ary trees—the parameter is described by explicit piecewise formulas with transitions determined by the relationship between depth and arity (Babu et al., 13 Dec 2025).

4. Graph Products and Structural Behavior

Mutual-visibility coloring interacts richly with graph products:

• For the lexicographic product $G \circ H$, if at least one factor is non-complete and $G$ is connected with $|G|\ge 2$, then $\chi_\mu(G \circ H) = 2$ (Brešar et al., 7 May 2025).
• For strong products and Cartesian products, sharp lower and upper bounds exist, often involving domination parameters or packing numbers (Babu et al., 11 Oct 2025, Brešar et al., 7 May 2025):

$$\chi_{\mu_2}(G \square H) \ge \max\{\chi_{\mu_2}(G) \rho_2(H), \chi_{\mu_2}(H) \rho_2(G)\}.$$

For strong products, $$\chi_{\mu_k}(G \boxtimes H) \leq \chi_{\mu_k}(G)\chi_{\mu_k}(H)$$.

In Hamming graphs (products of complete graphs), determining $\chi_\mu$ is open beyond specific cases such as Cartesian products of two cliques (Klavžar et al., 2024). For hypercubes $Q_n$, it is verified that

$\chi_\mu(Q_n) = \Theta(\log\log n),$

disproving conjectures that it may be bounded by a constant (Axenovich et al., 2024).

5. Algorithmic Aspects and Related Parameters

Finding a maximum mutual-visibility set is NP-hard, hence greedy coloring algorithms may fail to achieve optimal colorings in general (Klavžar et al., 2024). However, specialized structures—block graphs, strong grids, certain product graphs—admit efficient and sometimes optimal greedy solutions.

The mutual-visibility chromatic number is bounded above by several related parameters: in graphs of diameter two, $\chi_\mu(G) \leq \chi(G)$, and more generally, for regular graphs of large girth,

$\chi_\mu(G) \leq \lceil (n(G) - r^2 + 4)/2 \rceil$

holds for $r$-regular $G$ with girth $>6$ (Klavžar et al., 2024).

In the $k$-distance mutual-visibility generalization ($\chi_{\mu_k}(G)$), the classical clique-cover ($k=1$) and mutual-visibility chromatic number ($k \geq \text{diam}(G)$) appear as extremal cases (Babu et al., 11 Oct 2025). For $k=2$, tight bounds in terms of domination and total domination numbers have been established.

6. Relation to General Position and Ramsey Theory

The mutual-visibility chromatic number connects structurally to other combinatorial invariants, e.g.:

• General position chromatic number $\chi_{gp}(G)$.
• Domino parameters: domination number $\gamma(G)$, total domination number $\gamma_t(G)$.
• In subdivisions of complete graphs $S(K_n)$, $\chi_\mu(S(K_n))$ is tightly linked to Ramsey numbers $R(4;k)$: the minimum number $p(n)$ so that every edge-coloring of $K_n$ with $p(n)$ colors has a monochromatic $K_4$ (Brešar et al., 7 May 2025). This demonstrates Ramsey-theoretic intractability in exact computation.

In exact-distance graphs and corona constructions, new bounds and equivalences have emerged, notably $$\chi_\mu(G) \leq \chi_\mu(G \circ H) \leq \chi_\mu(G) + 1$$ (Klavžar et al., 2024).

7. Open Questions and Future Research Directions

Several open problems are fundamental:

• Determining $\chi_\mu(G)$ for families such as hypercubes, Hamming graphs, Sierpiński graphs, and product graphs in general (Babu et al., 13 Dec 2025, Axenovich et al., 2024, Klavžar et al., 2024).
• Full characterization of graphs with $\chi_\mu(G)=2$ (Klavžar et al., 2024).
• Determining the computational complexity in broader settings, with multiple conjectures regarding NP-completeness (Brešar et al., 7 May 2025).
• Study extremal bounds and efficient approximation algorithms in sparse graph classes (planar, chordal) and links to zero-forcing and general position numbers (Babu et al., 13 Dec 2025).

The interplay of mutual-visibility colorings with graph products, domination, partitioning, and Ramsey theory marks this parameter as central to emerging research in combinatorial optimization, extremal graph theory, and algorithmic complexity.