On stability of the Hamiltonian index under contractions and closures

The hamiltonian index of a graph $G$ is the smallest integer $k$ such that the $k$-th iterated line graph of $G$ is hamiltonian. We first show that, with one exceptional case, adding an edge to a graph cannot increase its hamiltonian index. We use this result to prove that neither the contraction of an $A_G(F)$-contractible subgraph $F$ of a graph $G$ nor the closure operation performed on $G$ (if $G$ is claw-free) affects the value of the hamiltonian index of a graph $G$

The Hamiltonian index of a graph and its branch-bonds

Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound. \u

The Hamiltonian index of graphs

The Hamiltonian index of a graph G is defined as h ( G ) = min { m : L m ( G ) is Hamiltonian } . In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44], we constructed a graph H ̃ ( m ) ( G ) from G and prove that if h ( G ) ≥ 2 , then h ( G ) = min{ m : H ̃ ( m ) ( G ) has a spanning Eulerian subgraph }

On Generalizations of Supereulerian Graphs

A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let $\kappa\u27(G)$ and $\delta(G)$ be the edge-connectivity and the minimum degree of a graph $G$, respectively. For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. This dissertation is devoted to providing some results on $(s,t)$-supereulerian graphs and supereulerian hypergraphs. In Chapter 2, we determine the value of the smallest integer $j(s,t)$ such that every $j(s,t)$-edge-connected graph is $(s,t)$-supereulerian as follows: j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right. As applications, we characterize $(s,t)$-supereulerian graphs when $t \ge 3$ in terms of edge-connectivities, and show that when $t \ge 3$, $(s,t)$-supereulerianicity is polynomially determinable. In Chapter 3, for a subset $Y \subseteq E(G)$ with $|Y|\le \kappa\u27(G)-1$, a necessary and sufficient condition for $G-Y$ to be a contractible configuration for supereulerianicity is obtained. We also characterize the $(s,t)$-supereulerianicity of $G$ when $s+t\le \kappa\u27(G)$. These results are applied to show that if $G$ is $(s,t)$-supereulerian with $\kappa\u27(G)=\delta(G)\ge 3$, then for any permutation $\alpha$ on the vertex set $V(G)$, the permutation graph $\alpha(G)$ is $(s,t)$-supereulerian if and only if $s+t\le \kappa\u27(G)$. For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Let $i_{s,t}(G)$ and $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $(s,t)$-supereulerian and $s$-Hamiltonian, respectively. In Chapter 4, for a simple graph $G$, we establish upper bounds for $i_{s,t}(G)$ and $h_s(G)$. Specifically, the upper bound for the $s$-Hamiltonian index $h_s(G)$ sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785]. Harary and Nash-Williams in 1968 proved that the line graph of a graph $G$ is Hamiltonian if and only if $G$ has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity. Applying the adjacency matrix of a hypergraph $H$ defined by Rodr\\u27iguez in 2002, let $\lambda_2(H)$ be the second largest adjacency eigenvalue of $H$. In Chapter 6, we prove that for an integer $k$ and a $r$-uniform hypergraph $H$ of order $n$ with $r\ge 4$ even, the minimum degree $\delta\ge k\ge 2$ and $k\neq r+2$, if $\lambda_2(H)\le (r-1)\delta-\frac{r^2(k-1)n}{4(r+1)(n-r-1)}$, then $H$ is $k$-edge-connected. %$\kappa\u27(H)\ge k$. Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The $(s,t)$-supereulerianicity of hypergraphs is another interesting topic to be investigated in the future

On Eulerian subgraphs and hamiltonian line graphs

A graph {\color{black}$G$} is Hamilton-connected if for any pair of distinct vertices {\color{black}$u, v \in V(G)$}, {\color{black}$G$} has a spanning $(u,v)$-path; {\color{black}$G$} is 1-hamiltonian if for any vertex subset $S \subseteq {\color{black}V(G)}$ with $|S| \le 1$, $G - S$ has a spanning cycle. Let $\delta(G)$, $\alpha\u27(G)$ and $L(G)$ denote the minimum degree, the matching number and the line graph of a graph $G$, respectively. The following result is obtained. {\color{black} Let $G$ be a simple graph} with $|E(G)| \ge 3$. If $\delta(G) \geq \alpha\u27(G)$, then each of the following holds. \\ (i) $L(G)$ is Hamilton-connected if and only if $\kappa(L(G))\ge 3$. \\ (ii) $L(G)$ is 1-hamiltonian if and only if $\kappa(L(G))\ge 3$. %==========sp For a graph $G$, an integer $s \ge 0$ and distinct vertices $u, v \in V(G)$, an $(s; u, v)$-path-system of $G$ is a subgraph $H$ consisting of $s$ internally disjoint $(u,v)$-paths. The spanning connectivity $\kappa^*(G)$ is the largest integer $s$ such that for any $k$ with $0 \le k \le s$ and for any $u, v \in V(G)$ with $u \neq v$, $G$ has a spanning $(k; u,v)$-path-system. It is known that $\kappa^*(G) \le \kappa(G)$, and determining if $\kappa^*(G) \u3e 0$ is an NP-complete problem. A graph $G$ is maximally spanning connected if $\kappa^*(G) = \kappa(G)$. Let $msc(G)$ and $s_k(G)$ be the smallest integers $m$ and $m\u27$ such that $L^m(G)$ is maximally spanning connected and $\kappa^*(L^{m\u27}(G)) \ge k$, respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for $msc(G)$ and $s_k(G)$, and characterized the extremal graphs reaching the upper bounds. %==============st For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is $(0,0)$-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph $G$ on $n$ vertices with $\delta(G) \ge \frac{n}{5} -1$, when $n$ is sufficiently large, is $(0,0)$-supereulerian or is contractible to $K_{2,3}$. We prove the following for any nonnegative integers $s$ and $t$. \\ (i) For any real numbers $a$ and $b$ with $0 \u3c a \u3c 1$, there exists a family of finitely many graphs \F(a,b;s,t) such that if $G$ is a simple graph on $n$ vertices with $\kappa\u27(G) \ge t+2$ and $\delta(G) \ge an + b$, then either $G$ is $(s,t)$-supereulerian, or $G$ is contractible to a member in \F(a,b;s,t). \\ (ii) Let $\ell K_2$ denote the connected loopless graph with two vertices and $\ell$ parallel edges. If $G$ is a simple graph on $n$ vertices with $\kappa\u27(G) \ge t+2$ and $\delta(G) \ge \frac{n}{2}-1$, then when $n$ is sufficiently large, either $G$ is $(s,t)$-supereulerian, or for some integer $j$ with $t+2 \le j \le s+t$, $G$ is contractible to a $j K_2$. %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: $\kappa\u27(G) \ge a$ and $\delta(G) \ge b\}$, and proposed a few problems to determine \cp(a,b) with $b \ge a \ge 4$ when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let $ess\u27(G)$ denote the essential edge-connectivity of a graph $G$, and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: $ess\u27(G) \ge a$ and $\delta(G) \ge b\}$. We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of $b \ge 1$, \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid

On the s-Hamiltonian index of a graph

In modeling communication networks by graphs, the problem of designing s-fault-tolerant networks becomes the search for s-Hamiltonian graphs. This thesis is a study of the s-Hamiltonian index of a graph G.;A path P of G is called an arc in G if all the internal vertices of P are divalent vertices of G. We define l (G) = max{lcub}m : G has an arc of length m that is not both of length 2 and in a K3{rcub}. We show that if a connected graph G is not a path, a cycle or K1,3, then for a given s, we give the best known bound of the s-Hamiltonian index of the graph

k-ordered hamiltonicity of iterated line graphs

AbstractA graph G of order n is k-ordered hamiltonian, 2≤k≤n, if for every sequence v1,v2,…,vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1,v2,…,vk in this order. In this paper, we generalize two well-known theorems of Chartrand on hamiltonicity of iterated line graphs to k-ordered hamiltonicity. We prove that if Ln(G) is k-ordered hamiltonian and n is sufficiently large, then Ln+1(G) is (k+1)-ordered hamiltonian. Furthermore, for any connected graph G, which is not a path, cycle, or the claw K1,3, there exists an integer N′ such that LN′+(k−3)(G) is k-ordered hamiltonian for k≥3