Problem 50 in the Open Problems Project of the computational geometry community asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian.

vVe prove that if G is a 2-edge-connected graph of order n 2: 14 and max{d{u),d(v)} > n3!) for each pair of nonadjacent vertices u~ v of G. then G contains a spanning Eulerian subgraph and hence the line graph of G is Hamiltonian.

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we prove that a complete graph on 2m (m = 2) vertices K2m can be properly edge-colored with 2m − 1 colors in such a way that the edges of K2m can be partitioned into m multicolored isomorphic spanning trees.

Hanoi graphs Hn p model the Tower of Hanoi game with p pegs and n discs. Sierpiński graphs Sn p arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that Sn p embeds as a spanning subgraph into Hn p if and only if p is odd or, trivially, if n = 1. MSC: 05C60

The Small Subgraph Conditioning Method has been used to study the almost sure existence and the asymptotic distribution of the number of regular spanning subgraphs of various types in random d -regular graphs. In this paper we use the method to determine the asymptotic distribution of the number of 3-star factors in random d -regular graphs for d ≥ 4.

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph Kn that has no rainbow spanning subgraph with diameter at most t. We prove s(n, t) = (n−2 2 ) +1 for n, t ≥ 3, while s(n, 2) = (n−2 2 )

In this paper we examine the classes of graphs whose Kn-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of Kn, the Kn-complement of H is the graph Kn H which is obtained from Kn by removing the edges of H . Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of ...

We give a relatively short graph theoretic proof of result Jordán and Tanigawa that 4-connected which has spanning plane triangulation as proper subgraph is generically globally rigid in $${\mathbb {R}}^3$$ . Our based on new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity {R}}^d$$

For a connected labelled graph G, a spanning tree T is a connected and an acyclic subgraph that spans all vertices of G. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of G. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate...

Let G be an infinite 4-connected planar graph such that the deletion of any finite set of vertices from G results in exactly one infinite component. Dean et al proved that either G admits a radial net or a special subgraph of G admits a ladder net, and they used these nets to show that G contains a spanning 1-way infinite path. In this paper, we show that if G admits a radial net, then G also c...