We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6connected biplane graph. Furthermore, we sh...

LetG be a connected graph of order n, and suppose that n = ∑k i=1 ni, where n1, n2, . . . , nk are integers with at least two. A spanning subgraph is called a path-factor if each component of it is a path of order at least two. In [2], Chen et al. gave a degree sum condition for the existence of a path-factor consisting of paths of order n1, n2, . . . , nk. In this paper, for 2-connected graphs...

Throughout today’s lecture, we fix a connected graph G and a strict threshold weighting θ : E(G)→ [0, 2] (such that ∆θ(H) := |V (H)| − ∑ e∈E(H) θ(e) > 0 for all ∅ ⊂ H ⊂ G). Also throughout today’s lecture, let α be a uniform random element of [n]V (G) (independent of Xθ). We think of α as a random choice of vertices from each color class of G ↑n. For each subgraph H ⊆ G, we have the isomorphic ...

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two nite structures to a global winning strategy. As app...

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The g-extraconnectivity κg(G) of a simple connected graph G is the minimum cardinality of a subset of V (G), if any, whose deletion disconnects G in such a way that every remaining component has at least g vertices. In this paper, we determine κg(Qn) for n+2 ≤ g ≤ 2n, n ≥ 4, where Qn denotes the n-dimensional hypercube.

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x) − f(y)| : xy∈ E(G)} taken over all proper numberings f of G. The composition of two graphs G and H , written as G[H ], is the graph with vertex set V (G) × V (H) and with (u1; v1) is adjacent to (u2; v2) if either u1 is adjacent to u2 in G or u1 =u2 and v1 is adjacent to v2 in H . In this paper, we investigate the bandwidt...

In this paper we discuss about tenacity and its properties in stability calculation. We indicate relationships between tenacity and connectivity, tenacity and binding number, tenacity and toughness. We also give good lower and upper bounds for tenacity.

A topology τ on the vertices of a comparability graph G is said to be compatible with G if each subgraph H of G is graph-connected if and only if it is a connected subspace of (G,τ). In two previous papers we considered the problem of finding compatible topologies for a given comparability graph and we proved that the nonexistence of infinite paths was a necessary and sufficient condition for t...

In this paper we introduce the class of decomposable discrete sets and give a polynomial algorithm for reconstructing discrete sets of this class from four projections. It is also shown that the class of decomposable discrete sets is more general than the class S ′ 8 of hv-convex 8but not 4-connected discrete sets which was studied in [3]. As a consequence we also get that the reconstruction fr...