Odd-sum colorings of planar graphs

Abstract

A coloring of a graph G is a map f:V(G)→Z+ such that f(v)≠f(w) for all vw∈E(G). A coloring f is an odd-sum coloring if ∑w∈N[v]f(w) is odd, for each vertex v∈V(G). The odd-sum chromatic number of a graph G, denoted χos(G), is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of G. Caro, Petruševski, and Škrekovski showed, among other results, that χos(G) is well-defined for every finite graph G and , in fact, χos(G)≤2χ(G). Thus, χos(G)≤8 for every planar graph G (by the 4 Color Theorem), χos(G)≤6 for every triangle-free planar graph G (by Grötzsch’s Theorem), and χos(G)≤4 for every bipartite graph.Caro et al. asked, for every even Δ≥4, whether there exists gΔ such that if G is planar with maximum degree Δ and girth at least gΔ then χos(G)≤5. They also asked, for every even Δ≥4, whether there exists gΔ such that if G is planar and bipartite with maximum degree Δ and girth at least gΔ then χos(G)≤3. We answer both questions negatively. We also refute a conjecture they made, resolve one further problem they posed, and make progress on another.