On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Complexity of Token Swapping and its Variants

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

Claw -free graphs and line graphs

The research of my dissertation is motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Tutte that every 4-edge-connected graph has a no-where-zero 3-flow. Towards the hamiltonian line graph problem, we proved that every 3-connected N2-locally connected claw-free graph is hamiltonian, which was conjectured by Ryjacek in 1990; that every 4-connected line graph of an almost claw free graph is hamiltonian connected, and that every triangularly connected claw-free graph G with |E( G)| ≥ 3 is vertex pancyclic. Towards the second conjecture, we proved that every line graph of a 4-edge-connected graph is Z 3-connected

A study on supereulerian digraphs and spanning trails in digraphs

A strong digraph D is eulerian if for any v ∈ V (D), d+D (v) = d−D (v). A digraph D is supereulerian if D contains a spanning eulerian subdigraph, or equivalently, a spanning closed directed trail. A digraph D is trailable if D has a spanning directed trail. This dissertation focuses on a study of trailable digraphs and supereulerian digraphs from the following aspects. 1. Strong Trail-Connected, Supereulerian and Trailable Digraphs. For a digraph D, D is trailable digraph if D has a spanning trail. A digraph D is strongly trail- connected if for any two vertices u and v of D, D posses both a spanning (u, v)-trail and a spanning (v,u)-trail. As the case when u = v is possible, every strongly trail-connected digraph is also su- pereulerian. Let D be a digraph. Let S(D) = {e ∈ A(D) : e is symmetric in D}. A digraph D is symmetric if A(D) = S(D). The symmetric core of D, denoted by J(D), has vertex set V (D) and arc set S(D). We have found a well-characterized digraph family D each of whose members does not have a spanning trail with its underlying graph spanned by a K2,n−2 such that for any strong digraph D with its matching number α′(D) and arc-strong-connectivity λ(D), if n = |V (D)| ≥ 3 and λ(D) ≥ α′(D) − 1, then each of the following holds. (i) There exists a family D of well-characterized digraphs such that for any digraph D with α′(D) ≤ 2, D has a spanning trial if and only if D is not a member in D. (ii) If α′(D) ≥ 3, then D has a spanning trail. (iii) If α′(D) ≥ 3 and n ≥ 2α′(D) + 3, then D is supereulerian. (iv) If λ(D) ≥ α′(D) ≥ 4 and n ≥ 2α′(D) + 3, then for any pair of vertices u and v of D, D contains a spanning (u, v)-trail. 2. Supereulerian Digraph Strong Products. A cycle vertex cover of a digraph D is a collection of directed cycles in D such that every vertex in D lies in at least one dicycle in this collection, and such that the union of the arc sets of these directed cycles induce a connected subdigraph of D. A subdigraph F of a digraph D is a circulation if for every vertex v in F, the indegree of v equals its outdegree, and a spanning circulation if F is a cycle factor. Define f(D) to be the smallest cardinality of a cycle vertex cover of the digraph D/F obtained from D by contracting all arcs in F , among all circulations F of D. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C′ with |C′| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. We prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f(D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian

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

Unavoidable minors in graphs and matroids

It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges to be contracted when they meet a degree-2 vertex. Dually, a parallel minor allows for any edge contractions but restricts the deletion of edges to those that lie in 2-edge cycles. This dissertation begins by proving the dual results to those noted above. These identify all unavoidable parallel minors for finite graphs of low connectivity. Following this, corresponding results on unavoidable minors for infinite graphs are proved. The dissertation concludes by finding the unavoidable parallel minors for 3-connected regular matroids, which combines the results for unavoidable series and parallel minors for graphs with Seymour\u27s decomposition theorem for regular matroids