The complexity of strong conflict-free vertex-connection $k$-colorability

TL;DR

This work introduces and analyzes the strong conflict-free vertex-connection coloring ($svcfc$), a variant requiring that every pair of vertices be connected by a conflict-free shortest path under a vertex coloring. It shows that for fixed $k\ge4$, $k$-$svcfc$ is NP-hard, with ETH-based $2^{o(n)}$ lower bounds on diameter-bounded graphs, and proves NP-hardness for $k=3$ even on graphs with diameter $3$, radius $2$, and domination number $3$. The authors also provide polynomial-time algorithms for optimal colorings on split graphs and co-bipartite graphs, establishing both hardness and tractable cases. The results highlight the stronger complexity of $3$-svcfc in restricted settings compared to classical $3$-coloring and introduce effective verification methods via level graphs, enriching the landscape of conflict-free connectivity problems.

Abstract

We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be strongly conflict-free vertex-connection $k$-colorable if $G$ admits a vertex $k$-coloring such that any two distinct vertices of $G$ are connected by a conflict-free $shortest$ path. Among others, we show that deciding whether a given graph is strongly conflict-free vertex-connection $3$-colorable is NP-complete even when restricted to $3$-colorable graphs with diameter $3$, radius $2$ and domination number $3$, and, assuming the Exponential Time Hypothesis (ETH), cannot be solved in $2^{o(n)}$ time on such restricted input graphs with $n$ vertices. This hardness result is quite strong when compared to the ordinary $3$-COLORING problem: it is known that $3$-COLORING is solvable in polynomial time in graphs with bounded domination number, and assuming ETH, cannot be solved in $2^{o(\sqrt{n})}$ time in $n$-vertex graphs with diameter $3$ and radius $2$. On the positive side, we point out that a strong conflict-free vertex-connection coloring with minimum color number of a given split graph or a co-bipartite graph can be computed in polynomial time.

Figures (2)

• Figure 1: The graph $Q_4$ (left) and $R_4$ (right).
• Figure 2: The graph $G$ obtained from the formula $\phi$ with $C_1=\{\overline{x},y,z\}$, $C_2=\{x,\overline{y},z\}$, $C_3=\{x,y,\overline{z}\}$ and $C_4=\{\overline{x},\overline{y},\overline{z}\}$.

Theorems & Definitions (24)

• definition 1: Strong conflict-free vertex-connection coloring
• proposition 1
• proposition 2
• proposition 3
• proof
• theorem 1
• corollary 1
• proof
• theorem 2
• proof