We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor 8(29+23 √ 2). Enroute to this, we also show that the Yao graph Y ∞ 4 in the L ∞ metric is a planar spanner with stretch factor 8.In the appendix, we improve the stretch factor and show that, in fact, Y k is a spanner for any k ≥ 7. Recently, Molla [4] showed that Y 2 and Y 3 are not
In 1990, Hendry Conjectured that every Hamiltonian chordal graph is cycle extendable; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on n vertices for any n ≥ 15. Furthermore, we show that there exist counterexamples where the ratio of the length of a nonextendable cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendability in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably Pn and the bull. Introduction.All graphs considered here are simple, finite, and undirected. A graph is Hamiltonian if it has a cycle containing all vertices; such a cycle is a Hamiltonian cycle. A graph G on n vertices is pancyclic if G contains a cycle of length m for every integer 3 ≤ m ≤ n. Let C and C be cycles in G of length m and m + 1, respectively, such that V (C ) \ V (C) = {v}. We say that C is an extension of C and that C is extendable (or, C extends through v to C ). If every non-Hamiltonian cycle of G is extendable, then G is cycle extendable. If, in addition, every vertex of G is contained in a triangle, then G is fully cycle extendable. The study of pancyclic graphs was initiated by Bondy [3], who recognized that most of the sufficient conditions for Hamiltonicity known at the time in fact implied a more complex cycle structure. Hendry [12] introduced the concept of cycle extendability and proved that many known sufficient conditions for a graph to be pancyclic in fact were sufficient for a graph to be (fully) cycle extendable.Given a graph G and a set of vertices U ⊆ V (G), we denote by G[U ] the subgraph obtained by deleting from G all vertices except those in U ; G[U ] is the subgraph induced by U , and a subgraph of G is an induced subgraph if it is induced by some U ⊆ V (G). A graph is chordal if it contains no induced cycles of length 4 or greater. It is not hard to show that every Hamiltonian chordal graph is pancyclic (see Proposition 3.4); however, the question of whether not every Hamiltonian chordal graph is cycle extendable has remained open since 1990.Conjecture 1.1 (Hendry's Conjecture [12]). If G is a Hamiltonian chordal graph, then G is fully cycle extendable.In this paper, we settle Hendry's Conjecture in the negative. In section 2, we show that (a) for any n ≥ 15 there exists a counterexample to Hendry's Conjecture on n vertices and (b) for every real number α > 0 there exists a counterexample G with a nonextendable cycle C such that |V (C)| < α|V (G)|. The question then remains: *
Abstract:A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen.
We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor [Formula: see text]. Enroute to this, we also show that the Yao graph [Formula: see text] in the L∞ metric is a plane spanner with stretch factor 8.
An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yield a proper vertex colouring. If such an assignment from a set $S$ exists, we say the graph is $S$-weight colourable. We consider the $S$-weight colourability of digraphs by defining the accumulated weight at a vertex to be the sum of the inbound weights minus the sum of the outbound weights. Bartnicki et al. showed that every digraph is $S$-weight colourable for any set $S$ of size $2$ and asked whether one could show the same result using an algebraic approach. Using the Combinatorial Nullstellensatz and a classical theorem of Schur, we provide such a solution.
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