Jill R. Faudree

On k-ordered graphs

We prove a hypergraph version of Halls theorem. The proof is topological.

Forbidden subgraphs that imply 2-factors

The connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph is hamiltonian have been characterized by Bedrossian [Forbidden subgraph and minimum degree conditions for hamiltonicity, Ph.D. Thesis, Memphis State University, 1991], and extensions of these excluding graphs for general graphs of order at least 10 were proved by Faudree and Gould [Characterizing forbidden pairs for Hamiltonian properties, Discrete Math. 173 (1997) 45-60]. In this paper a complete characterization of connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph of order at least 10 has a 2-factor will be proved. In particular it will be shown that the characterization for 2-factors is very similar to that for hamiltonian cycles, except there are seven additional pairs. In the case of graphs of all possible orders, there are four additional forbidden pairs not in the hamiltonian characterization, but a claw is part of each pair.

Forbidden subgraphs that imply k -ordered and k -ordered Hamiltonian

For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices in the sequence in the given order. If the cycle is also a hamiltonian cycle, then G is said to be k-ordered hamiltonian. Forbidden connected subgraphs and forbidden pairs of connected subgraphs that imply that a 2-connected graph is hamiltonian have been characterized. Each of these forbidden subgraph conditions will be investigated to determine if it implies more than just hamiltonicity, but in fact it implies k-ordered or k-ordered hamiltonian in the presence of the appropriate connectivity on the graph. More general classes of forbidden subgraphs that imply k-ordered and k-ordered hamiltonian will also be considered.

On the Non-(p−1)-Partite Kp-Free Graphs

Abstract We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?

On Forbidden Pairs Implying Hamilton‐Connectedness

Let X, Y be connected graphs. A graph G is -free if G contains a copy of neither X nor Y as an induced subgraph. Pairs of connected graphs such that every 3-connected -free graph is Hamilton connected have been investigated most recently in (Guantao Chen and Ronald J. Gould, Bull. Inst. Combin. Appl., 29 (2000), 25–32.) [8] and (H. Broersma, R. J. Faudree, A. Huck, H. Trommel, and H. J. Veldman, J. Graph Theory, 40(2) (2002), 104–119.) [5]. This paper improves those results. Specifically, it is shown that every 3-connected -free graph is Hamilton connected for and or N1, 2, 2 and the proof of this result uses a new closure technique developed by the third and fourth authors. A discussion of restrictions on the nature of the graph Y is also included.

Highly Incident Configurations with Chiral Symmetry

A geometric

Uniquely Tree-saturated Graphs

k

Degree sum and vertex dominating paths

-configuration is a collection of points and straight lines in the plane so that