On the additive chromatic number of several families of graphs

Abstract

Let f : V → {1, . . . , k} be a labeling of the vertices of a graph G = (V, E) and denote with f (N (v)) the sum of the labels of all vertices adjacent to v. The least value k for which a graph G admits a labeling satisfying f (N (u)) = f (N (v)) for all (u, v) ∈ E is called additive chromatic number of G and denoted η(G). It was first presented by Czerwiński, Grytczuk and Zelazny who also proposed a conjecture that for every graph G, η(G) ≤ χ(G), where χ(G) is the chromatic number of G. Bounds of η(G) are known for very few families of graphs. In this work, we show that the conjecture holds for split graphs by giving an upper bound of the additive chromatic number and we present exact formulas for computing η(G) when G is a fan, windmill, circuit, wheel, complete split, headless spider, cycle/wheel/complete sun, regular bipartite or complete multipartite observing that the conjecture is satisfied in all cases. In addition, we propose an integer programming formulation which is used for checking the conjecture over all connected graphs up to 10 vertices.

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“…The conjecture remains open, but some progress has been made-for example, it was shown in [6] that if G is planar, η(G) ≤ 468, and more results on the additive chromatic number of planar graphs can be found in [1], [7], and [9]. Progress for other types of graphs can be found in [10], where an up-to-date list of classes of graphs for which the conjecture has been verified is given. Since η(G) ≤ η ℓ (G) for every graph G, Theorem 4.2 offers a possible method for finding upper bounds on η(G), and therefore may be a relevant tool for helping to prove Czerwiński et al's conjecture for certain classes of graphs.…”