On dominating and spanning circuits in graphs

An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was made by Catlin, who defined the reduction of a graph G and showed that G has a spanning circuit if and only if the reduction of G has a spanning circuit. We refine Catlin's reduction technique to obtain a result which contains several known and new sufficient conditions for a graph to have a spanning or dominating circuit in terms of degree-sums of adjacent vertices. In particular, the result implies the truth of the following conjecture of Benhocine et al.: If G is a connected simple graph of order n such that every cut edge of G is incident with a vertex of degree 1 and d(u)+d(v) > 2(1/5n-1) for every edge uv of G, then, for n sufficiently large, L(G) is hamiltonian

Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved

Spanning closed trails in graphs

Let G be a 2-edge-connected simple graph on n\u3e;95 vertices. Let l be the number of vertices of degree 2 in G. We prove that if

Reductions of graphs and spanning Eulerian subgraphs

This dissertation is primarily focused on conditions for the existence of spanning closed trails in graphs. However, results in my dissertation and the method we used, which was invented by Catlin, are not only useful for finding spanning closed trails in graphs, but also useful to study double cycle cover problems, hamiltonian line graphs problems, and dominating closed trail problems, etc. A graph is called supereulerian if it contains a spanning closed trail. Several, people have worked on the conditions for spanning closed trails having the form d(u) + d(v) $\u3e$ cn (0 $\u3c$ c $\u3c$ 1) for all edge uv in graph, as in the paper of Benhocine, Clark, Kohler and Veldman (3), etc. When obtaining good sufficient conditions having the form d(u) + d(v) $\u3e$ cn , it is useful to show that a graph G is either supereulerian or it can be contracted to a nonsupereulerian graph G\sp\prime having a large matching. This was done. For example, we show that if a 3-edge-connected simple graph has no spanning closed trail then it can be contracted to a nonsupereulerian graph G\sp\prime of order n \sp\prime whose maximum matching has size at least (n\sp\prime + 4)/3, and it is best possible. By using this result, we show that if G is a 3-edge-connected simple graph of order n, if for every edge uv, d(u) + d(v) $\geq$ $n \over 5$ $-$ 2, then either G has a spanning closed trail of G is contractible to the Petersen graph. By a theorem of Harary and Nash-Williams, this implies that either the line graph L(G) is hamiltonian or G can be contracted to the Petersen graph. This proves a conjecture of Benhocine, Clark, Kohler and Veldmann (3) for 3-edge-connected graphs, and with a stronger conclusion. We also study the conditions for supereulerian graphs having form \sum\sbsp{i=1}{t} d(u\sb{i}) $\u3e$ cn (0 $\u3c$ c $\u3c$ 1) where t is a positive integer and no two vertices of \{ u\sb1,\ u\sb2,\cdots u\sb{t} \} are adjacent. We obtain some best possible results on this aspect which improve the results of Benhocine, Clark, Kohler, & Veldman (3), Calin (11), (13), (14), Clark (24), Z. Q. Chen & Y. F. Xue (23), Lesniak-Foster & Willianson (30) and Veldman (36) significantly. We also study the following extremal graph theory problem: For a family F of graphs and for a natural number n, what is the maximum size of simple graphs of order n which are not in F, where F = $\{$supereulerian graphs with clique number $m\}$. (Note that when F = $\{$graphs with clique number at least $m\}$, this is Turan\u27s Theorem.) One of our results proves a conjecture of Cai (10)