19 research outputs found
Ordered and linked chordal graphs
A graph G is called k-ordered if for every sequence of k distinct vertices there is a cycle traversing these vertices in the given order. In the present paper we consider two novel generalizations of this concept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered. We prove the following results for a chordal graph G: (a) G is (2k-3)-connected if and only if it is k-vertex-edge-ordered (k ≥ 3). (b) G is (2k-1)-connected if and only if it is strongly k-vertex-edge-ordered (k ≥ 2). (c) G is k-linked if and only if it is (2k-1)-connected
Cyclability, Connectivity and Circumference
In a graph , a subset of vertices is said to be
cyclable if there is a cycle containing the vertices in some order. is said
to be -cyclable if any subset of vertices is cyclable. If any
\textit{ordered} vertices are present in a common cycle in that order, then the
graph is said to be -ordered. We show that when ,
-cyclable graphs also have circumference , and that this is
best possible. Furthermore when , , and
for -ordered graphs we show . We also generalize a
result by Byer et al. on the maximum number of edges in nonhamiltonian
-connected graphs, and show that if is a -connected graph of order with , then the graph is
hamiltonian, and moreover the extremal graphs are unique
4-connected triangulations and 4-orderedness
AbstractFor a positive integer k≥4, a graph G is called k-ordered, if for any ordered set of k distinct vertices of G, G has a cycle that contains all the vertices in the designated order. Goddard (2002) [3] showed that every 4-connected triangulation of the plane is 4-ordered. In this paper, we improve this result; every 4-connected triangulation of any surface is 4-ordered. Our proof is much shorter than the proof by Goddard
k-ordered hamiltonicity of iterated line graphs
AbstractA graph G of order n is k-ordered hamiltonian, 2≤k≤n, if for every sequence v1,v2,…,vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1,v2,…,vk in this order. In this paper, we generalize two well-known theorems of Chartrand on hamiltonicity of iterated line graphs to k-ordered hamiltonicity. We prove that if Ln(G) is k-ordered hamiltonian and n is sufficiently large, then Ln+1(G) is (k+1)-ordered hamiltonian. Furthermore, for any connected graph G, which is not a path, cycle, or the claw K1,3, there exists an integer N′ such that LN′+(k−3)(G) is k-ordered hamiltonian for k≥3