arXiv subjects

Julien Bensmail

Publications and source records attributed to Julien Bensmail.

24 records · Page 2Linked to original sources

arXiv2015

On oriented cliques with respect to push operation

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices. We also provide an exhaustive list of minimal (with respect to spanning subgraph inclusion) planar push cliques.

math.CO↗

arXiv2015

List coloring digraphs

The dichromatic number $\vecχ(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's Conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as $Ω(\log_2 n)$. We finally give a Brooks-type upper bound on the list dichromatic number of digon-free digraphs.

math.CO↗

arXiv2015

Analogous to cliques for (m,n)-colored mixed graphs

Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $χ(G)$ of $G$ will be the order of the smallest complete graph to which $G$ admits a homomorphism to. As every graph, which is not a complete graph, admits a homomorphism to a smaller complete graph, we can redefine the chromatic number $χ(G)$ of $G$ to be the order of the smallest graph to which $G$ admits a homomorphism to. Of course, such a smallest graph must be a complete graph as they are the only graphs with chromatic number equal to their order. The concept of vertex coloring can be generalize for other types of graphs. Naturally, the chromatic number is defined to be the order of the smallest graph (of the same type) to which a graph admits homomorphism to. The analogous notion of clique turns out to be the graphs with order equal to their (so defined) "chromatic number". These "cliques" turns out to be much more complicated than their undirected counterpart and are interesting objects of study. In this article, we mainly study different aspects of "cliques" for signed (graphs with positive or negative signs assigned to each edge) and switchable signed graphs (equivalence class of signed graph with respect to switching signs of edges incident to the same vertex).

math.CO↗

arXiv2015

Strong edge-coloring of $(3, Δ)$-bipartite graphs

A strong edge-coloring of a graph $G$ is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most $3$ and the other part is of maximum degree $Δ$. For every such graph, we prove that a strong $4Δ$-edge-coloring can always be obtained. Together with a result of Steger and Yu, this result confirms a conjecture of Faudree, Gyárfás, Schelp and Tuza for this class of graphs.

cs.DM↗

arXiv2015

Orienting edges to fight fire in graphs

We investigate a new oriented variant of the Firefighter Problem. In the traditional Firefighter Problem, a fire breaks out at a given vertex of a graph, and at each time interval spreads to neighbouring vertices that have not been protected, while a constant number of vertices are protected at each time interval. In the version of the problem considered here, the firefighters are able to orient the edges of the graph before the fire breaks out, but the fire could start at any vertex. We consider this problem when played on a graph in one of several graph classes, and give upper and lower bounds on the number of vertices that can be saved. In particular, when one firefighter is available at each time interval, and the given graph is a complete graph, or a complete bipartite graph, we present firefighting strategies that are provably optimal. We also provide lower bounds on the number of vertices that can be saved as a function of the chromatic number, of the maximum degree, and of the treewidth of a graph. For a subcubic graph, we show that the firefighters can save all but two vertices, and this is best possible.

cs.DM↗

arXiv2014

Strong edge-colouring of sparse planar graphs

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $Δ$ has a strong edge-colouring with at most $4Δ+4$ colours. We show that $3Δ+1$ colours suffice if the graph has girth 6, and $4Δ$ colours suffice if $Δ\geq 7$ or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.

cs.DM↗