Algorithm on rainbow connection for maximal outerplanar graphs

TL;DR

This paper studies the rainbow connection number of maximal outerplanar graphs, providing conditions for equality with diameter, and introduces a linear-time algorithm for rainbow coloring based on the graph's structure.

Contribution

It presents a new polynomial-time algorithm for rainbow coloring MOPs with a bounded number of colors, improving understanding of their rainbow connection properties.

Findings

Conditions where rainbow connection number equals diameter.

Linear-time algorithm for rainbow coloring MOPs.

Bound on the number of colors used in rainbow coloring.

Abstract

In this paper, we consider rainbow connection number of maximal outerplanar graphs(MOPs) on algorithmic aspect. For the (MOP) $G$, we give sufficient conditions to guarantee that $rc(G) = diam(G).$ Moreover, we produce the graph with given diameter $d$ and give their rainbow coloring in linear time. X.Deng et al. $\cite{XD}$ give a polynomial time algorithm to compute the rainbow connection number of MOPs by the Maximal fan partition method, but only obtain a compact upper bound. J. Lauri $\cite{JL}$ proved that, for chordal outerplanar graphs given an edge-coloring, to verify whether it is rainbow connected is NP-complete under the coloring, it is so for MOPs. Therefore we construct Central-cut-spine of MOP $G,$ by which we design an algorithm to give a rainbow edge coloring with at most $2rad(G)+2+c,0\leq c\leq rad(G)-2$ colors in polynomial time.