Abstract
For a nontrivial connected graph G, let \({c: V(G)\to {{\mathbb N}}}\) be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N(v) denote the set of vertices adjacent to v. The color sum σ(v) of v is the sum of the colors of the vertices in N(v). If σ(u) ≠ σ(v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ(G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ(G) = a and χ(G) = b. There is a connected graph G of order n with σ(G) = k for every pair k, n of positive integers with k ≤n if and only if k ≠ n − 1. Several other results concerning sigma chromatic numbers are presented.
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References
Addario-Berry L., Aldred R.E.L., Dalal K., Reed B.A.: Vertex colouring edge partitions. J. Combin. Theory Ser. B 94, 237–244 (2005)
Chartrand G., Jacobson M.S., Lehel J., Oellermann O.R., Ruiz S., Saba F.: Irregular networks. Congr. Numer. 64, 197–210 (1988)
Chartrand G., Zhang P.: Chromatic Graph Theory. Chapman & Hall/CRC, Boca Raton (2008)
Goss, J.M., Fonger, N., Phillips, B., Segroves, C.N.: Research report, Western Michigan University (2009)
Kalkowski, Karoński, M., Pfender, F.: Vertex-coloring edge-weightings: towards the 1-2-3 Conjecture. J. Combin. Theory Ser. B. (to appear)
Karoński M., Łuczak T., Thomason A.: Edge weights and vertex colours. J. Combin. Theory Ser. B 91, 151–157 (2004)
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Chartrand, G., Okamoto, F. & Zhang, P. The Sigma Chromatic Number of a Graph. Graphs and Combinatorics 26, 755–773 (2010). https://doi.org/10.1007/s00373-010-0952-7
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DOI: https://doi.org/10.1007/s00373-010-0952-7