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

Factors and Connected Factors in Tough Graphs with High Isolated Toughness

In this paper, we show that every $1$-tough graph with order and isolated toughness at least $r+1$ has a factor whose degrees are $r$, except for at most one vertex with degree $r+1$. Using this result, we conclude that every $3$-tough graph with order and isolated toughness at least $r+1$ has a connected factor whose degrees lie in the set $\{r,r+1\}$, where $r\ge 3$. Also, we show that this factor can be found $m$-tree-connected, when $G$ is a $(2m+\epsilon)$-tough graph with order and isolated toughness at least $r+1$, where $r\ge (2m-1)(2m/\epsilon+1)$ and $\epsilon > 0$. Next, we prove that every $(m+\epsilon)$-tough graph of order at least $2m$ with high enough isolated toughness admits an $m$-tree-connected factor with maximum degree at most $2m+1$. From this result, we derive that every $(2+\epsilon)$-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set $\{2,4\}$. In addition, we provide a family of $5/3$-tough graphs with high enough isolated toughness having no connected even factors with bounded maximum degree

A note on interconnecting matchings in graphs

AbstractWe prove a sufficient condition for a graph G to have a matching that interconnects all the components of a disconnected spanning subgraph of G. The condition is derived from a recent extension of the Matroid intersection theorem due to Aharoni and Berger. We apply the result to the problem of the existence of a (spanning) 2-walk in sufficiently tough graphs

Spanning k-trees and distance spectral radius in graphs

Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $\lambda_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$ denotes the distance matrix of $G$. In this paper, we verify a upper bound for $\lambda_1(D(G))$ in a connected graph $G$ to guarantee the existence of a spanning $k$-tree in $G$.Comment: 11 page