arXiv subjects
Publications and source records attributed to Jean-Luc Fouquet.
Reed conjectured that for any graph $G$, $χ(G) \leq \lceil \frac{ω(G)+Δ(G)+1}{2}\rceil$, where $χ(G)$, $ω(G)$, and $Δ(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of $P_5$-free graphs or $Chair$-free graphs.
In 1998, Reed conjectured that for any graph $G$, $χ(G) \leq \lceil \frac{ω(G) + Δ(G)+1}{2}\rceil$, where $χ(G)$, $ω(G)$, and $Δ(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we study this conjecture for some expansions of graphs, that is graphs obtained with the well known operation composition of graphs. We prove that Reed's Conjecture holds for expansions of bipartite graphs, for expansions of odd holes where the minimum chromatic number of the components is even, when some component of the expansion has chromatic number 1 or when a component induces a bipartite graph. Moreover, Reed's Conjecture holds if all components have the same chromatic number, if the components have chromatic number at most 4 and when the odd hole has length 5. Finally, when $G$ is an odd hole expansion, we prove $χ(G)\leq\lceil\frac{ω(G)+Δ(G)+1}{2}\rceil+1$.
In a graph $G$ of maximum degree $Δ$ let $γ$ denote the largest fraction of edges that can be $Δ$ edge-coloured. Albertson and Haas showed that $γ\geq 13/15$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for which $γ= 13/15$. This extends a result given by Steffen. These results are obtained by using structural properties of the so called $δ$-minimum edge colourings for graphs with maximum degree 3. Keywords : Cubic graph; Edge-colouring
The notion of a $δ$-minimum edge-colouring was introduced by J-L. Fouquet (in his french PhD Thesis \cite{FouPhD}). Here we present some structural properties of $δ$-minimum edge-colourings, partially taken from the above thesis. The paper serves as an auxiliary tool for another paper submitted by the authors to Graphs and Combinatorics.
A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for each vertex. We examine this notion and show that a cubic 3 edge-colorable graph can always be provided with three compatible normal odd partitions. The Petersen graph has this property and we can construct other cubic graphs with chromatic index four with the same property. Finally, we propose a new conjecture which, if true, would imply the well known Fan and Raspaud Conjecture
In a graph $G$ of maximum degree 3, let $γ(G)$ denote the largest fraction of edges that can be 3 edge-coloured. Rizzi \cite{Riz09} showed that $γ(G) \geq 1-\frac{2\strut}{\strut 3 g_{odd}(G)}$ where $g_{odd}(G)$ is the odd girth of $G$, when $G$ is triangle-free. In \cite{FouVan10a} we extended that result to graph with maximum degree 3. We show here that $γ(G) \geq 1-\frac{2 \strut}{\strut 3 g_{odd}(G)+2}$, which leads to $γ(G) \geq 15/17$ when considering graphs with odd girth at least 5, distinct from the Petersen graph.
If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a{\em Fulkerson covering}) with the property that every edge of $G$ is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A {\em FR-triple} is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.
We consider the Seidel complementation on ($P_5, \bar{P_5}, Bull)$-free graphs
We consider cubic graphs formed with $k \geq 2$ disjoint claws $C_i \sim K_{1, 3}$ ($0 \leq i \leq k-1$) such that for every integer $i$ modulo $k$ the three vertices of degree 1 of $\ C_i$ are joined to the three vertices of degree 1 of $C_{i-1}$ and joined to the three vertices of degree 1 of $C_{i+1}$. Denote by $t_i$ the vertex of degree 3 of $C_i$ and by $T$ the set $\{t_1, t_2,..., t_{k-1}\}$. In such a way we construct three distinct graphs, namely $FS(1,k)$, $FS(2,k)$ and $FS(3,k)$. The graph $FS(j,k)$ ($j \in \{1, 2, 3\}$) is the graph where the set of vertices $\cup_{i=0}^{i=k-1}V(C_i) \setminus T$ induce $j$ cycles (note that the graphs $FS(2,2p+1)$, $p\geq2$, are the flower snarks defined by Isaacs \cite{Isa75}). We determine the number of perfect matchings of every $FS(j,k)$. A cubic graph $G$ is said to be {\em 2-factor hamiltonian} if every 2-factor of $G$ is a hamiltonian cycle. We characterize the graphs $FS(j,k)$ that are 2-factor hamiltonian (note that FS(1,3) is the "Triplex Graph" of Robertson, Seymour and Thomas \cite{RobSey}). A {\em strong matching} $M$ in a graph $G$ is a matching $M$ such that there is no edge of $E(G)$ connecting any two edges of $M$. A cubic graph having a perfect matching union of two strong matchings is said to be a {\em\Jaev}. We characterize the graphs $FS(j,k)$ that are \Jaesv.
A conjecture of Máucajová and uSkoviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a stronger result for traceable graphs.
If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings $M_1,...,M_6$ of $G$ with the property that every edge of $G$ is contained in exactly two of them and Berge conjectured that its edge set can be covered by 5 perfect matchings. We define $τ(G)$ as the least number of perfect matchings allowing to cover the edge set of a bridgeless cubic graph and we study this parameter. The set of graphs with perfect matching index 4 seems interesting and we give some informations on this class.
A conjecture of Fan and Raspaud [3] asserts that every bridgeless cubic graph con-tains three perfect matchings with empty intersection. Kaiser and Raspaud [6] sug-gested a possible approach to this problem based on the concept of a balanced join in an embedded graph. We give here some new results concerning this conjecture and prove that a minimum counterexample must have at least 32 vertices.
A normal partition of the edges of a cubic graph is a partition into trails (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition. We investigate this notion and give some results and problems.
We extend the notion of $P_4$-sparse graphs previously introduced by {\scshape Hoàng} by considering $\mathcal{F}$-sparse graphs were $\mathcal{F}$ denotes a finite set of graphs on $p$ vertices. Thus we obtain some results on $(P_5,\bar{P_5})$-sparse graphs already known on $(P_5,\bar{P_5})$-free graphs. Finally we completely describe the structure of $(P_5,\bar{P_5}, bull$)-sparse graphs, it follows that those graphs have bounded clique-width.