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

Edge-dominating cycles, k-walks and Hamilton prisms in 2K22K_2-free graphs

We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.Comment: LaTeX, 8 page

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Hammer et al. and Merris, the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure

Spanning trees with many leaves: new extremal results and an improved FPT algorithm

We present two lower bounds for the maximum number of leaves in a spanning tree of a graph. For connected graphs without triangles, with minimum degree at least three, we show that a spanning tree with at least (n+4)/3 leaves exists, where n is the number of vertices of the graph. For connected graphs with minimum degree at least three, that contain D diamonds induced by vertices of degree three (a diamond is a K4 minus one edge), we show that a spanning tree exists with at least (2n-D+12)/7 leaves. The proofs use the fact that spanning trees with many leaves correspond to small connected dominating sets. Both of these bounds are best possible for their respective graph classes. For both bounds simple polynomial time algorithms are given that find spanning trees satisfying the bounds. \ud \ud The second bound is used to find a new fastest FPT algorithm for the Max-Leaf Spanning Tree problem. This problem asks whether a graph G on n vertices has a spanning tree with at least k leaves. The time complexity of our algorithm is f(k)g(n), where g(n) is a polynomial, and f(k) Î O(8.12k).\ud \ud \u