Parity and Strong Parity Edge-Coloring of Graphs

A parity walk in an edge-coloring of a graph is a walk traversing each color an even number of times. We introduce two parameters. Let p(G) be the least number of colors in a parity edge-coloring of G (a coloring having no parity path). Let b p(G) be the least number of colors in a strong parity edge-coloring of G (a coloring having no open parity walk). Note that b p(G) ≥ p(G) ≥ χ′(G). The values p(G) and b p(G) may be equal or differ, with equality conjectured for all bipartite graphs. If G is connected, then p(G) ≥ dlg |V (G)|e, with equality for paths and even cycles (Cn needs one more color for odd n). The proof that b p(Kn) = 2dlg ne − 1 for all n will appear later; the conjecture that p(Kn) = b p(Kn) is proved here for n ≤ 16 and other cases. Also, p(K2,n) = b p(K2,n) = 2 dn/2e. In general, b p(Km,n) ≤ m′ dn/m′e, where m′ = 2dlg me. We compare these to other parameters and pose many open questions.