arXiv subjects
Publications and source records attributed to Marek Skrzypczyk.
The quadratic embedding property of graphs consisting of three paths (theta graphs) is fully characterised. For this aim, a theorem by Winkler (1985) is utilized. An alternative proof of that result using the RKHS technique is presented.
We show that for any integer $k \ge 4$, every oriented graph with minimum semidegree bigger than $\frac{1}{2}(k-1+\sqrt{k-3})$ contains an antidirected path of length $k$. Consequently, every oriented graph on $n$ vertices with more than $(k-1+\sqrt{k-3})n$ edges contains an antidirected path of length $k$. This asymptotically proves the antidirected path version of a conjecture of Stein and of a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomassé, respectively.
We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points and common endpoints. As a result, we provide an infinite family of primary graphs which are not quadratically embeddable.