TY  - JOUR
A1  - Chandran, Sunil L.
A1  - Issac, Davis
A1  - Lauri, Juho
A1  - van Leeuwen, Erik Jan
T1  - Upper bounding rainbow connection number by forest number
JF  - Discrete mathematics
N2  - A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G).& nbsp;A simple way to rainbow-connect a graph G is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of G. This proves that rc(G) <= |V (G)|-1. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of t(G)-1 for rc(G), where t(G) is the number of vertices in the largest induced tree of G? The answer turns out to be negative, as there are counter-examples that show that even c .t(G) is not an upper bound for rc(G) for any given constant c.& nbsp;In this work we show that if we consider the forest number f(G), the number of vertices in a maximum induced forest of G, instead of t(G), then surprisingly we do get an upper bound. More specifically, we prove that rc(G) <= f(G) + 2. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound.
KW  - rainbow connection
KW  - forest number
KW  - upper bound
Y1  - 2022
U6  - https://doi.org/10.1016/j.disc.2022.112829
SN  - 0012-365X
SN  - 1872-681X
VL  - 345
IS  - 7
PB  - Elsevier
CY  - Amsterdam [u.a.]
ER  -