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 $15$

Spanning Eulerian subgraphs and Catlin’s reduced graphs

A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following: (i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large; (ii) the size of a maximum matching in G is at most 6; (iii) the independence number of G is at most 5. These are improvements of prior results in [16], [18], [24] and [25]

Eulerian subgraphs containing given vertices and hamiltonian line graphs

AbstractLet G be a graph and let D1(G) be the set of vertices of degree 1 in G. Veldman (1994) proves the following conjecture from Benhocine et al. (1986) that if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), d(x) + d(y) > (2n)/5 − 2, then for n large, L(G), the line graph of G, is hamiltonian. We shall show the following improvement of this theorem: if G − D1(G) is a 2-edge-connected simple graph with n vertices and if for every edge xy ∈ E(G), max[;d(x), d(y)] ⩾ n/5 − 1, then for n large, L(G) is hamiltonian with the exception of a class of well characterized graphs. Our result implies Veldman's theorem

Spanning Trails and Spanning Trees

There are two major parts in my dissertation. One is based on spanning trail, the other one is comparing spanning tree packing and covering.;The results of the spanning trail in my dissertation are motivated by Thomassen\u27s Conjecture that every 4-connected line graph is hamiltonian. Harary and Nash-Williams showed that the line graph L( G) is hamiltonian if and only if the graph G has a dominating eulerian subgraph. Also, motivated by the Chinese Postman Problem, Boesch et al. introduced supereulerian graphs which contain spanning closed trails. In the spanning trail part of my dissertation, I proved some results based on supereulerian graphs and, a more general case, spanning trails.;Let alpha(G), alpha\u27(G), kappa( G) and kappa\u27(G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. First, we discuss the 3-edge-connected graphs with bounded edge-cuts of size 3, and prove that any 3-edge-connected graph with at most 11 edge cuts of size 3 is supereulerian, which improves Catlin\u27s result. Second, having the idea from Chvatal-Erdos Theorem which states that every graph G with kappa(G) ≥ alpha( G) is hamiltonian, we find families of finite graphs F 1 and F2 such that if a connected graph G satisfies kappa\u27(G) ≥ alpha(G) -- 1 (resp. kappa\u27(G) ≥ 3 and alpha\u27( G) ≤ 7), then G has a spanning closed trail if and only if G is not contractible to a member of F1 (resp. F2). Third, by solving a conjecture posed in [Discrete Math. 306 (2006) 87-98], we prove if G is essentially 4-edge-connected, then for any edge subset X0 ⊆ E(G) with |X0| ≤ 3 and any distinct edges e, e\u27 2 ∈ E(G), G has a spanning ( e, e\u27)-trail containing all edges in X0.;The results on spanning trees in my dissertation concern spanning tree packing and covering. We find a characterization of spanning tree packing and covering based on degree sequence. Let tau(G) be the maximum number of edge-disjoint spanning trees in G, a(G) be the minimum number of spanning trees whose union covers E(G). We prove that, given a graphic sequence d = (d1, d2···dn) (d1 ≥ d2 ≥···≥ dn) and integers k2 ≥ k1 \u3e 0, there exists a simple graph G with degree sequence d satisfying k 1 ≤ tau(G) ≤ a(G) ≤ k2 if and only if dn ≥ k1 and 2k1(n -- 1) ≤ Sigmani =1 di ≤ 2k2( n -- 1 |I| -- 1) + 2Sigma i∈I di, where I = {lcub}i : di \u3c k2{rcub}

Small cycle cover, group coloring with related problems

Bondy conjectured that if G is a simple 2-connected graph with n ≥ 3 vertices, then the edges of G can be covered by at most 2n-33 cycles. In Chapter 2, a result on small cycle cover is obtained and we also show that the result is as best as possible.;Thomassen conjectured that every 4-connected line graph is hamiltonian. In Chapters 3 and 4, we apply Catlin\u27s reduction method to study cycles in line graphs. Results about hamiltonian connectivity of line graphs and 3-edge-connected graphs are obtained. Several former results are extended.;Jaeger, Linial, Payan and Tarsi introduced group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters 5 and 6 are devoted to the study of group coloring of graphs.;The concept of list coloring, choosability and choice number was introduced by Erdos, Rubin and Taylor in 1979 and Vizing in 1976. Alon and Tarsi proved that every bipartite planar graph is 3-choosable. Thomassen showed that every planar graph is 5-choosable and that every planar graph with girth at least 5 is 3-choosable. In Chapter 7, results on list coloring are obtained, extending a former result of Thomassen

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