Henry Liu

Total rainbow k-connection in graphs

Abstract Let k be a positive integer and G be a k -connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k -connection number r c k ( G ) of G . An edge-coloured path is rainbow if its edges have distinct colours. Then, r c k ( G ) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function r c k ( G ) has since been studied by numerous researchers. An analogue of the function r c k ( G ) involving vertex colourings, the rainbow vertex k -connection number r v c k ( G ) , was subsequently introduced. In this paper, we introduce a version which involves total colourings. A total-coloured path is total-rainbow if its edges and internal vertices have distinct colours. The total rainbow k -connection number of G , denoted by t r c k ( G ) , is the minimum number of colours required to colour the edges and vertices of G , so that any two vertices of G are connected by k internally vertex-disjoint total-rainbow paths. We study the function t r c k ( G ) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions r c k ( G ) , r v c k ( G ) , and t r c k ( G ) , by considering how close and how far apart t r c k ( G ) can be from r c k ( G ) and r v c k ( G ) .

Rainbow vertex connection of digraphs

An edge-coloured path is rainbow if its edges have distinct colours. An edge-coloured connected graph is said to be rainbow connected if any two vertices are connected by a rainbow path, and strongly rainbow connected if any two vertices are connected by a rainbow geodesic. The (strong) rainbow connection number of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). As an extension, Krivelevich and Yuster proposed the concept of rainbow vertex-connection. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.

The Balanced Decomposition Number and Vertex Connectivity

The balanced decomposition number

Monochromatic Kr-Decompositions of Graphs

f(G)

Rainbow k-connection in Dense Graphs (Extended Abstract)

of a graph

Decompositions of Graphs into Fans and Single Edges

G

Monochromatic Clique Decompositions of Graphs

was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced coloring of a graph

The Balanced Decomposition Number of TK4 and Series-Parallel Graphs

G

Rainbow connection for some families of hypergraphs

is a coloring of some of the vertices of

Monochromatic K r -Decompositions of Graphs

G