Local gap colorings from edge labelings

We study a local version of gap vertex-distinguishing edge coloring. From an edge labeling f : E(G) → {1, . . . , k} of a graph G, an induced vertex coloring c is obtained by coloring the vertices with the greatest difference between incident edge labels. The local gap chromatic number χ∆(G) is ∗ Partially funded by NSF GK-12 Transforming Experiences Grant DGE-0742434. † Partially funded by Simons Foundation grant #276726. A. BRANDT ET AL. /AUSTRALAS. J. COMBIN. 65 (3) (2016), 200–211 201 the minimum k for which there exists an edge coloring such that c(u) 6= c(v) for all edges uv. We prove that χ(G) ≤ χ∆(G) ≤ χ(G) + 1 for all graphs G, where χ(G) denotes the chromatic number of G. Further, we provide graph classes attaining both bounds.