On rainbow total-coloring of a graph
A path P P connecting two vertices u u and v v in a total-colored graph G G is called a rainbow total-path between u u and v v if all elements in V(P)?E(P) V ( P ) ? E ( P ) , except for u u and v v , are assigned distinct colors. The total-colored graph G G is rainbow total-connected if it has a rainbow total-path between every two vertices. The rainbow total-connection number, denoted by rtc(G) r t c ( G ) , of a graph G G is the minimum colors such that G G is rainbow total-connected. It was shown that rtc(G)?m(G)+n?(G) r t c ( G ) ? m ( G ) + n ? ( G ) , and the equality holds if and only if G G is a tree, where n?(G) n ? ( G ) is the number of inner vertices of G G . In this paper, we show that rtc(G)?m(G)+n?(G)?1,m(G)+n?(G)?2 r t c ( G ) ? m ( G ) + n ? ( G ) ? 1 , m ( G ) + n ? ( G ) ? 2 and characterize the graphs with rtc(G)=m(G)+n?(G)?3 r t c ( G ) = m ( G ) + n ? ( G ) ? 3 . With this result, the following sharp upper bound holds: for a connected graph G G , if G G is not a tree, then rtc(G)?m(G)+n?(G)?3 r t c ( G ) ? m ( G ) + n ? ( G ) ? 3 ; moreover, the equality holds if and only if G G belongs to five graph classes. We also investigate the NordhausGaddum-type lower bounds for the rainbow total-connection number of a graph and derive that if G G is a connected graph of order n?8 n ? 8 , then r t c ( G ) + r t c ( G  ) ? 6 and r t c ( G ) r t c ( G  ) ? 9 . An example is given to show that both of these bounds are sharp.
rainbow total-coloring rainbow total-connection number nordhausgaddum-type