Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

Abstract

A path in an edge-colored graph $G$ is rainbow if no two edges of it arecolored the same. The graph $G$ is rainbow-connected if there is a rainbow pathbetween every pair of vertices. If there is a rainbow shortest path betweenevery pair of vertices, the graph $G$ is strongly rainbow-connected. Theminimum number of colors needed to make $G$ rainbow-connected is known as therainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly,the minimum number of colors needed to make $G$ strongly rainbow-connected isknown as the strong rainbow connection number of $G$, and is denoted by$\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether$\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclassof chordal graphs. Furthermore, there exists no polynomial-time algorithm forapproximating the strong rainbow connection number of an $n$-vertex split graphwith a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. Wethen turn our attention to block graphs, which also form a subclass of chordalgraphs. We determine the strong rainbow connection number of block graphs, andshow it can be computed in linear time. Finally, we provide a polynomial-timecharacterization of bridgeless block graphs with rainbow connection number atmost 4.Comment: 13 pages, 3 figure