ANTI-RAMSEY NUMBER OF EDGE-DISJOINT RAINBOW SPANNING TREES IN ALL GRAPHS

An edge-colored graph H is called rainbow if every edge of H receives a different color. Given any host multigraph G, the anti-Ramsey number of t edge-disjoint rainbow spanning trees in G, denoted by r(G, t), is defined as the maximum number of colors in an edge-coloring of G containing no t edge-disjoint rainbow spanning trees. For any vertex partition P, let E(P,G) be the set of noncrossing edges in G with respect to P. In this paper, we determine r(G, t) for all host multigraphs G: r(G, t) = IE(G)I if there exists a partition P0 with IE(G)I IE(P0, G)I < t(IP0I 1); and r(G, t) = maxP : | P | \geq3{IE(P,G)I + t(IPI 2)\} otherwise. As a corollary, we determine r(Kp,q, t) for all values of p, q, t, improving a result of Jia, Lu, and Zhang.