Supereulerian Properties in Graphs and Hamiltonian Properties in Line Graphs

Following the trend initiated by Chvatal and Erdos, using the relation of independence number and connectivity as sufficient conditions for hamiltonicity of graphs, we characterize supereulerian graphs with small matching number, which implies a characterization of hamiltonian claw-free graph with small independence number.;We also investigate strongly spanning trailable graphs and their applications to hamiltonian connected line graphs characterizations for small strongly spanning trailable graphs and strongly spanning trailable graphs with short longest cycles are obtained. In particular, we have found a graph family F of reduced nonsupereulerian graphs such that for any graph G with kappa\u27(G) ≥ 2 and alpha\u27( G) ≤ 3, G is supereulerian if and only if the reduction of G is not in F..;We proved that any connected graph G with at most 12 vertices, at most one vertex of degree 2 and without vertices of degree 1 is either supereulerian or its reduction is one of six exceptional cases. This is applied to show that if a 3-edge-connected graph has the property that every pair of edges is joined by a longest path of length at most 8, then G is strongly spanning trailable if and only if G is not the wagner graph.;Using charge and discharge method, we prove that every 3-connected, essentially 10-connected line graph is hamiltonian connected. We also provide a unified treatment with short proofs for several former results by Fujisawa and Ota in [20], by Kaiser et al in [24], and by Pfender in [40]. New sufficient conditions for hamiltonian claw-free graphs are also obtained

Hamiltonicity of 3-connected line graphs

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. conjectured [H. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571-576] that every 3-connected, essentially 4-connected line graph is Hamiltonian. In this note, we first show that the conjecture posed by Lai et al. is not true and there is an infinite family of counterexamples; we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is Hamiltonian; examples show that all conditions are sharp. (C) 2012 Elsevier Ltd. All rights reserved.NSFC [11171279, 11071016]; Beijing Natural Science Foundation [1102015

On 3-connected hamiltonian line graphs

AbstractThomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and Vumar (2009) [5] found another family of graphs, called P3D graphs, which properly contain all quasi-claw-free graphs. In this paper, we investigate the hamiltonicity of 3-connected line graphs of DCT graphs and P3D graphs, and prove that if G is a DCT graph or a P3D graph with κ(L(G))≥3 and if L(G) does not have an independent vertex 3-cut, then L(G) is hamiltonian. Consequently, every 4-connected line graph of a DCT graph or a P3D graph is hamiltonian

Graphes super-eulériens, problèmes hamiltonicité et extrémaux dans les graphes

In this thesis, we focus on the following topics: supereulerian graphs, hamiltonian line graphs, fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees, and several extremal problems on the (minimum and/or maximum) size of graphs under a given graph property. The thesis includes six chapters. The first one is to introduce definitions and summary the main results of the thesis, and in the last chapter we introduce the furture research of the thesis. The main studies in Chapters 2 - 5 are as follows. In Chapter 2, we explore conditions for a graph to be supereulerian.In Section 1 of Chapter 2, we characterize the graphs with minimum degree at least 2 and matching number at most 3. By using the characterization, we strengthen the result in [93] and we also address a conjecture in the paper.In Section 2 of Chapter 2, we prove that if d(x)+d(y)≥ n-1-p(n) for any edge xy∈E(G), then G is collapsible except for several special graphs, where p(n)=0 for n even and p(n)=1 for n odd. As a corollary, a characterization for graphs satisfying d(x)+d(y)≥ n-1-p(n) for any edge xy∈E(G) to be supereulerian is obtained. This result extends the result in [21]. In Section 3 of Chapter 2, we focus on a conjecture posed by Chen and Lai [Conjecture~8.6 of [33]] that every 3-edge connected and essentially 6-edge connected graph is collapsible. We find a kind of sufficient conditions for a 3-edge connected graph to be collapsible. In Chapter 3, we mainly consider the hamiltonicity of 3-connected line graphs. In the first section of Chapter 3, we give several conditions for a line graph to be hamiltonian, especially we show that every 3-connected, essentially 11-connected line graph is hamilton- connected which strengthens the result in [91]. In the second section of Chapter 3, we show that every 3-connected, essentially 10-connected line graph is hamiltonian-connected. In the third section of Chapter 3, we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is hamiltonian. Moreover, if G has 10 vertices of degree 3 and its line graph is not hamiltonian, then G can be contractible to the Petersen graph.In Chapter 4, we consider edge fault-tolerant hamiltonicity of Cayley graphs generated by transposition trees. We first show that for any F ⊆ E(Cay(B:S_{n})), if |F| ≤ n-3 and n ≥ 4, then there exists a hamiltonian path in Cay(B:S_{n})-F between every pair of vertices which are in different partite sets. Furthermore, we strengthen the above result in the second section by showing that Cay(S_n,B)-F is bipancyclic if Cay(S_n,B) is not a star graph, n ≥ 4 and |F| ≤ n-3. In Chapter 5, we consider several extremal problems on the size of graphs. In Section 1 of Chapter 5, we bounds the size of the subgraph induced by m vertices of hypercubes. We show that a subgraph induced by m (denote m by ∑_{i=0}^{s}2^{t_i}, t₀=[log₂m] and t_i=[log₂(m-(∑_{r=0}^{i-1}2^{t_r}))] for i ≥ 1) vertices of an n-cube (hypercube) has at most ∑_{i=0}^{s}t_{i}2^{t_i-1} + ∑_{i=0}^{s} i...2^{t_i} edges. As its applications, we determine the m-extra edge-connectivity of hypercubes for m ≤ 2^{n/2} and g-extra edge-connectivity of the folded hypercube for g ≤ n.In Section 2 of Chapter 5, we partially study the minimum size of graphs with a given minimum degree and a given edge degree. As an application, we characterize some kinds of minimumrestricted edge connected graphs. In Section 3 of Chapter 5, we consider the minimum size of graphs satisfying Ore-condition.Dans cette thèse, nous concentrons sur les sujets suivants: super-eulérien graphe, hamiltonien ligne graphes, le tolerant aux pannes hamiltonien laceabilité de Cayley graphe généré par des transposition arbres et plusieurs problèmes extrémaux concernant la (minimum et/ou maximum) taille des graphes qui ont la même propriété.Cette thèse comprend six chapitres. Le premier chapitre introduit des définitions et indique la conclusion des resultants principaux de cette thèse, et dans le dernier chapitre, nous introduisons la recherche de furture de la thèse. Les travaux principaux sont montrés dans les chapitres 2-5 comme suit:Dans le chapitre 2, nous explorons les conditions pour qu'un graphe soit super-eulérien.Dans la section 1, nous caractérisons des graphes dont le dégrée minimum est au moins de 2 et le nombre de matching est au plus de 3. Dans la section 2, nous prouvons que si pour tous les arcs xy∈E(G), d(x)+d(y)≥ n-1-p(n), alors G est collapsible sauf quelques bien définis graphes qui ont la propriété p(n)=0 quand n est impair et p(n)=1 quand n est pair. Dans la section 3 de la Chapitre 2, nous trouvons les conditions suffisantes pour que un graphe de 3-arcs connectés soit pliable. Dans le chapitre 3, nous considérons surtout l'hamiltonien de 3-connecté ligne graphe. Dans la première section de Chapitre 3, nous montrons que chaque 3-connecté, essentiellement 11-connecté ligne graphe est hamiltonien-connecté. Cela renforce le résultat dans [91]. Dans la seconde section de Chapitre 3, nous montrons que chaque 3-connecté, essentiellement 10-connecté ligne graphe est hamiltonien-connecté. Dans la troisième section de Chapitre 3, nous montrons que 3-connecté, essentiellement 4-connecté ligne graphe venant d'un graphe qui comprend au plus 9 sommets de degré 3 est hamiltonien. Dans le chapitre 4, nous montrons d'abord que pour tous F ⊆ E(Cay(B:S_{n})), si |F| ≤ n-3 et n ≥ 4, il existe un hamiltonien graphe dans Cay(B:S_{n})-F entre tous les paires de sommets qui sont dans les différents partite ensembles. De plus, nous renforçons le résultat figurant ci-dessus dans la seconde section montrant que Cay(S_n,B)-F est bipancyclique si Cay(S_n,B) n'est pas un star graphe, n ≥ 4 et |F| ≤ n-3. Dans le chapitre 5, nous considérons plusieurs problems extrémaux concernant la taille des graphes. Dans la section 1 de Chapitre 5, nous bornons la taille de sous-graphe provoqué par m sommets de hypercubes (n-cubes). Dans la section 2 de Chapitre 5, nous étudions partiellement la taille minimale d'un graphe savant son degré minimum et son degré d'arc. Dans la section 3 de Chapitre 5, nous considérons la taille minimale des graphes satisfaisants la Ore-condition