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 .
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References
Bollobás, B.: Extremal Graph Theory. New York-London: Academic Press: 1978
Kubicka, E.: private communication through Schwenk A.J.
Kubicka, E., Schwenk, A.J.: An introduction to chromatic sums. In: Proc. ACM Computer Science Conference, Louisville (Kentucky) 1989, pp. 39–45.
Thomassen, C., Erdös, P., Alavi, Y., Malde, P.J., Schwenk, A.J.: Tight bounds on the chromatic sum of a connected graph. J. Graph Theory13, 353–357 (1989)
Tuza, Zs.: Problems and results on graph and hypergraph colorings. Le Matematiche (to appear)
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Tuza, Z. Contractions and minimalk-colorability. Graphs and Combinatorics 6, 51–59 (1990). https://doi.org/10.1007/BF01787480
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DOI: https://doi.org/10.1007/BF01787480