arXiv subjects
Publications and source records attributed to Isaac Wass.
We describe symbolic constructions for listing and enumerating graphs having the same induced edge label sequence. We settle in the affirmative R. Whitty's [W08] conjectured existence of determinantal constructions for listing and enumerating gracefully labeled trees. We conclude the paper with a description of a new graceful labeling algorithm.
An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds on the Ramsey number of ordered matchings arising from sum-decomposable permutations, an alternating ordering of the cycle, and an alternating ordering of the tight hyperpath. We also construct ordered matchings on $n$ vertices whose Ramsey number is $n^{q+o(1)}$ for any given exponent $q\in(1,2)$.