A dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. A result of Broersma from 1988 implies that if G is an n-vertex k-connected graph and δ(G) > n−k k+2 − 1, then G contains a dominating path. We prove the following results. The lengths of dominating paths include all values from the shortest up to at least min{n − 1,2δ(G)}. For δ(G) > an, where a is a...

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, minimum chromatic index of respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ overfull $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since matching in can have size at most $\lfloor \rfloor$, it follows that $\chi'(G) = \Delta...

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors wn associated to the vertices of the tree and depending only on their individual degrees n. We focus on the case when wn grows faster than exponentially with n. In this case the measures on trees of finite size N converge weakly as N tends to infinity to a measure which is co...

Recently a number of variants of the approximate minimum degree algorithm have been proposed that aim to efficiently order symmetric matrices containing some dense rows. We compare the performance of these variants on a range of problems and highlight their limitations. This leads us to propose a new variant that offers both speed and robustness.