Hexagonal Tilings and Locally C6 Graphs

We give a complete classification of hexagonal tilings and locally C6 graphs, by showing that each of them has a natural embedding in the torus or in the Klein bottle. We also show that locally grid graphs are minors of hexagonal tilings (and by duality of locally C6 graphs) by contraction of a perfect matching and deletion of the resulting parallel edges, in a form suitable for the study of their Tutte uniqueness.Comment: 14 figure

A note on perfect orders

AbstractPerfectly orderable graphs were introduced by Chvátal in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplical vertex, in the graph or in its complement. Second, weprovide a characterization of graphs G with this property: each maximal vertex ofG is simplical in the complement of G. Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order

On the quasi-locally paw-free graphs

AbstractIn this paper, we present a new class of graphs named quasi-locally paw-free (QLP) graphs. We prove the strong perfect graph conjecture for a subclass of QLP class, by exhibiting a polynomial combinatorial algorithm for ω-coloring any Berge graph for this subclass. This subclass contains K4-free graphs and chordal graphs

Phase transition for random walks on graphs with added weighted random matching

For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider, for various sequences of graphs $(G_n)$ and corresponding weights $(\varepsilon_n)$, whether the (weighted) random walk on $(G_n^*)$ has cutoff. In particular, we show a phase transition for two families of graphs, graphs with polynomial growth of balls, and graphs where the entropy of the simple random walk grows linearly up to time of order $\log|V_n|$. These include in particular tori, expander families and locally expanding families

On the separability of graphs

Recently, Cicalese and Milanič introduced a graph-theoretic concept called separability. A graph is said to be k-separable if any two non-adjacent vertices can be separated by the removal of at most k vertices. The separability of a graph G is the least k for which G is k-separable. In this paper, we investigate this concept under the following three aspects. First, we characterize the graphs for which in any non-complete connected induced subgraph the connectivity equals the separability, so-called separability-perfect graphs. We list the minimal forbidden induced subgraphs of this condition and derive a complete description of the separability-perfect graphs.We then turn our attention to graphs for which the separability is given locally by the maximum intersection of the neighborhoods of any two non-adjacent vertices. We prove that all (house,hole)-free graphs fulfill this property – a class properly including the chordal graphs and the distance-hereditary graphs. We conclude that the separability can be computed in O(m∆) time for such graphs.In the last part we introduce the concept of edge-separability, in analogy to edge-connectivity, and prove that the class of k-edge-separable graphs is closed under topological minors for any k. We explicitly give the forbidden topological minors of the k-edge-separable graphs for each 0 ≤ k ≤ 3