<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Liu, H.</style></author><author><style face="normal" font="default" size="100%">Mestre, Â.</style></author><author><style face="normal" font="default" size="100%">Sousa, Teresa</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Total rainbow k-connection in graphs</style></title><secondary-title><style face="normal" font="default" size="100%">Discrete Applied Mathematics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/tmjs/files/trc-preprint.pdf</style></url></related-urls></urls><volume><style face="normal" font="default" size="100%">174</style></volume><pages><style face="normal" font="default" size="100%">92-101</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;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 rc_k(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rc_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 rc_k(G) has since been studied by numerous researchers. An analogue of the function rc_k(G) involving vertex colourings, the rainbow vertex k-connection number rvc_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 trc_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 trc_k(G) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions rc_k(G), rvc_k(G), and trc_k(G), by considering how close and how far apart trc_k(G) can be from rc_k(G) and rvc_k(G).&lt;/p&gt;
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