CYCLES CONTAINING SPECIFIED EDGES IN A GRAPH

The aim of this paper is to prove that if s > 1 and G is a graph of order n > 4s + 6 satisfying 2 > (4n - 4s - 3) / 3 ; then every matching of G lies on a cycle of length at least n-s and hence, in a path of length at least n - s + 1

GRAPHS WITH EVERY PATH OF LENGTH k IN A HAMILTONIAN CYCLE

In this paper we prove that if G is a (k + 2)-connected graph on n > 3 vertices satisfying P(n + k) : dG(x; y) = 2 ) maxfd(x); d(y)g > n + k 2 for each pair of vertices x and y in G; then any path S G of length k is contained in a hamiltonian cycle of G

Cyclability in bipartite graphs

Let $G=(X,Y,E)$ be a balanced $2$-connected bipartite graph and $S \subset V(G)$. We will say that $S$ is cyclable in $G$ if all vertices of $S$ belong to a common cycle in $G$. We give sufficient degree conditions in a balanced bipartite graph $G$ and a subset $S \subset V(G)$ for the cyclability of the set $S$

Cyclability in bipartite graphs

Let $G=(X,Y,E)$ be a balanced $2$-connected bipartite graph and $S \subset V(G)$. We will say that $S$ is cyclable in $G$ if all vertices of $S$ belong to a common cycle in $G$. We give sufficient degree conditions in a balanced bipartite graph $G$ and a subset $S \subset V(G)$ for the cyclability of the set $S$

A degree condition implying that every matching is contained in a hamiltonian cycle

AbstractWe give a degree sum condition for three independent vertices under which every matching of a graph lies in a hamiltonian cycle. We also show that the bound for the degree sum is almost the best possible

Problèmes extrémaux en théorie des graphes, généralisations du problème hamiltonien

La description de la théorie extrémale des graphes donnée par B. Bollobásest la suivante :La théorie extrémale des graphes est une branche de la théorie des graphes qui consiste à trouver des relations entre divers invariants de graphes comme l'ordre, les degrés maximum et minimum, le minimum de la somme des degrés pris pour toute paire de sommets non adjacents. Plus généralement, à trouver les valeurs extrêmes de ces invariants qui assurent qu'un graphe possède une propriété donnée.Dans un graphe G un cycle qui contient tous les sommets de G est un cycle hamiltonien de G, Un graphe est hamiltonien si et seulement s'il contient un cycle hamiltonien.En général, le problème hamiltonien est un problème qui consiste à trouver des conditions suffisantes pour qu'un graphe possède un chemin hamiltonien ou un cycle hamiltonien. C'est l'un des problèmes les plus connus en théorie des graphes.Deux des résultats les plus connus de ce type sont le résultat de O. Ore sur la somme des degrés des sommets indépendants et le résultat de G. Fan sur le maximum des degrés des sommets de distance 2. Nous donnons quelques généralisations de ces deux theorèmes. Avec des conditions portants sur les degrés des sommets nous obtenons des cycles hamiltoniens ou non, contenant un couplage ou des cycles contenant un ensemble donné des sommets. Les résultats sont obtenus pour n'importe quel graphe ou dans le cas particulier des graphes bipartis. Dans certains cas, nous donnons des exemples qui montrent que les résultats obtenus sont les meilleurs possibles.The description of extremal graph theory given by B. Bollob\'as is the following:Extremal graph theory is a branch of graph theory concerning with finding relations between various graph invariants like order, minimal and maximal degrees, minimal sum of degrees of nonadjacent vertices. More generally, with finding the extremal values of these invariants ensuring that a graph has a given property. Note that many problems in graph theory could be formulated as extremal problems.In a graph G a cycle that contains every vertex of G is called a hamiltonian cycle of G. A graph is Hamiltonian iff it contains a hamiltonian cycle.In general the hamiltonian problem is a problem concerning with finding sufficient conditions under which the graph has a hamiltonian path or cycle. It is one of the most known problems in graph theory.Two of the most known results of this type are the result of O. Ore with the sum of degrees of independent vertices and the result of G. Fan with the maximum of degrees of vertices of distance two.We give some generalization of these results. Under conditions on the sum of degrees of independent vertices we obtain a cycle or a hamiltonian cycle containing a set of independent edges or a cycle containing a given set of vertices. The obtained results are for arbitrary graphs or for the special case of bipartite graphs. In some cases we give examples showing that the results are best possible.ORSAY-PARIS 11-BU Sciences (914712101) / SudocSudocFranceF