S. F. Kapoor

n-Hamiltonian graphs

Abstract A graph G with p≥3 points, 0≤n≤p−3, is called n-Hamiltonian if the removal of any k points from G, 0≤k≤n, results in a Hamiltonian graph. This generalizes the concept of Hamiltonian graphs in as much as the 0-Hamiltonian graphs are precisely the Hamiltonian graphs. Sufficient conditions for a graph to be n-Hamiltonian are presented, including generalizations of results on Hamiltonian graphs due to Dirac, Ore, and Posa.

GRAPHS WITH PRESCRIBED DEGREE SETS AND GIRTH

For a finite nonempty set of integers, each of which is at least two, and an integern ≥ 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.

ON HOMOGENEOUSLY TRACEABLE NONHAMILTONIAN GRAPHS

A graph G is homogeneously traceable if for every vertex v of G there exists a hamiltonian path with initial vertex v. It is shown that there exists a homogeneously traceable nonhamiltonian graph of order p, for all positive integers p except for 3 ≤p≤ 8. Further, if G is a homogeneously traceable nonhamiltonian graph of order p, then {5p/4} is a sharp lower bound on the size of G and the maximum degree of G is at most p– 4 for p≥ 9.

Hamiltonian path graphs

The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.

The partial complement of graphs

AbstractFor a graphG, the switched graphSv(G) ofG at a vertexv is the graph obtained fromG by deleting the edges ofG incident withv and adding the edges of

On randomly 3-axial graphs

\bar G

A sufficient condition for n -connectedness of graphs

incident withv. Properties of graphs whereSv(G) ≅G or