Strong conflict-free connection of graphs

A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of the edges of $P$. An edge-colored graph $G$ is called \emph{conflict-free connected} if for each pair of distinct vertices of $G$ there is a conflict-free path in $G$ connecting them. The graph $G$ is called \emph{strongly conflict-free connected }if for every pair of vertices $u$ and $v$ of $G$ there exists a conflict-free path of length $d_G(u,v)$ in $G$ connecting them. For a connected graph $G$, the \emph{strong conflict-free connection number} of $G$, denoted by $\mathit{scfc}(G)$, is defined as the smallest number of colors that are required in order to make $G$ strongly conflict-free connected. In this paper, we first show that if $G_t$ is a connected graph with $m$ $(m\geq 2)$ edges and $t$ edge-disjoint triangles, then $\mathit{scfc}(G_t)\leq m-2t$, and the equality holds if and only if $G_t\cong S_{m,t}$. Then we characterize the graphs $G$ with $scfc(G)=k$ for $k\in \{1,m-3,m-2,m-1,m\}$. In the end, we present a complete characterization for the cubic graphs $G$ with $scfc(G)=2$.Comment: 23 pages, 10 figure

Hardness results for three kinds of colored connections of graphs

The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers.Comment: 23 pages, 7 figure

Proper-walk connection number of graphs

This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with $k$ colours for every possible value of $k$.Comment: 25 pages, 9 figure