Solution of the Hamiltonian problem for self-complementary graphs (original) (raw)

Hamiltonian-connected self-complementary graphs

Discrete Mathematics, 1994

A self-complementary graph having a complementing permutation d = [ 1,2,3, ,4k], consisting of one cycle, and having the edges (1,2) and (1,3) is strongly Hamiltonian iff it has an edge between two even-labelled vertices. Some of these strongly Hamiltonian self-complementary graphs are also shown to be Hamiltonian connected.

On almost self-complementary graphs

Discrete Mathematics, 2006

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. Almost self-complementary circulant graphs were first studied by Dobson andŠajna in 2004. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost selfcomplementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.

The degree sequences of self-complementary graphs

Journal of Combinatorial Theory, Series B, 1976

Known necessary conditions for realization of a sequence of integers as the degrees of a self-complementary graph are shown to he sufficient. An algorithm for constructing a realization of such a sequence as degrees of such a graph is illustrated by an example.

More on almost self-complementary graphs

Discrete Mathematics, 2009

A graph X is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. If I is a perfect matching in X c and ϕ : X → X c − I is an isomorphism, then the graph X is said to be fairly almost self-complementary if ϕ preserves I setwise, and unfairly almost self-complementary if it does not. In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary. Two perfect matchings I and J in X c are said to be X-non-isomorphic if no isomorphism from X + I to X + J induces an automorphism of X. We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X-nonisomorphic perfect matchings for every even order greater than or equal to four.

Almost self-complementary circulant graphs

Discrete Mathematics, 2004

An almost self-complementary graph is a graph isomorphic to its complement minus a 1-factor. An almost self-complementary circulant graph is called cyclically almost self-complementary if it has an isomorphic almost complement with the same regular cyclic subgroup of the automorphism group. In this paper we prove that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-complementary circulants. We also describe the structure of cyclically almost self-complementary circulants and the action of their automorphism groups. Finally, we exhibit a class of almost self-complementary Cayley graphs on a dihedral group that are isomorphic to cyclically almost self-complementary circulants.

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

2014

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac's Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore's condition is another early result giving a sufficient condition for Hamiltonicity...

New sufficient conditions for hamiltonian and pancyclic graphs

Discussiones Mathematicae Graph Theory, 2007

For a graph G of order n we consider the unique partition of its vertex set V (G) = A ∪ B with A = {v ∈ V (G) : d(v) ≥ n/2} and B = {v ∈ V (G) : d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.

D Equienergetic Self Complementary Graphs - 2009

The D-eigenvalues {µ 1 , µ 2 , . . . , µ n } of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by spec D (G) . The D-energy E D (G) of the graph G is the sum of the absolute values of its D-eigenvalues. We describe here the distance spectrum of some self-complementary graphs in the terms of their adjacency spectrum. These results are used to show that there exists D-equienergetic self-complementary graphs of order n = 48t and 24(2t + 1) for t ≥ 4 .