Contractions and minimalk-colorability

Abstract

Coloring the vertex set of a graphG with positive integers, thechromatic sum Σ(G) ofG is the minimum sum of colors in a proper coloring. Thestrength ofG is the largest integer that occurs in every coloring whose total isΣ(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strengths has at least ((2 +$\sqrt 2$)^s−1 − (2 −$\sqrt 2$)^s−1)/$\sqrt 2$ vertices (s ≥ 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for everys ≥ 3 there exist precisely two treesT _s andR _s such that every tree of strength at leasts is edge-contractible toT _s orR _s .