Rainbow connection numbers and the minimum degree sum of a graph

Abstract

Let <italic>G</italic> be a connected graph with order <italic>n</italic> and minimum degree <italic>δ</italic>. The existing upper bounds of the rainbow connection number and the rainbow vertex-connection number in terms of the minimum degree are very large, linear in <italic>n</italic>, if <italic>δ</italic> is small and <italic>n</italic> is large. In this paper, we want to replace <italic>δ</italic> by another parameter <italic>σ_k</italic>+<italic>k</italic>, the minimum degree sum of <italic>k</italic> independent vertices, to improve the upper bounds signi cantly. We prove that if <italic>G</italic> is a connected graph of order <italic>n</italic> with <italic>k</italic> independent vertices, then rc(<italic>G</italic>)≤3<italic>kn</italic>/(<italic>σ_k</italic>+<italic>k</italic>)+6<italic>k</italic>-3. We also prove that rvc(<italic>G</italic>)≤(4<italic>k</italic>+2<italic>k</italic>²)<italic>n</italic>/(<italic>σ_k</italic>+<italic>k</italic>)+5<italic>k</italic> if <italic>σ_k</italic>≤7<italic>k</italic> or <italic>σ_k</italic>≥8<italic>k</italic>; whereas rvc(<italic>G</italic>)≤(38<italic>k</italic>/9+2<italic>k</italic>²)<italic>n</italic>/(<italic>σ_k</italic>+<italic>k</italic>)+5<italic>k</italic> if 7<italic>k</italic><<italic>k</italic><8<italic>k</italic>. Examples are given to show that our bounds are much better than the existing ones, i.e., for <italic>δ</italic> which is very small but <italic>σ_k</italic> is very large, and our bounds are rc(<italic>G</italic>)≤9<italic>k</italic>-3 and rvc(<italic>G</italic>)≤9<italic>k</italic>+2<italic>k</italic>² or rvc(<italic>G</italic>)≤(83<italic>k</italic>/9)+2<italic>k</italic>², which imply that rc(<italic>G</italic>) and rvc(<italic>G</italic>) can be bounded by constants from our upper bounds, but linear in <italic>n</italic> from the existing ones.