Summary: Siberian Mathematical Journal, Vol. 35, No. 3, 1994
DOMINATING SETS AND HAMILTONICITY
IN K 1;3 ­FREE GRAPHS y)
A. A. Ageev UDC 519.17
1. Introduction. An undirected graph is called K 1;3 ­free, if it contains no induced subgraph
isomorphic to the complete bipartite graph K 1;3 . The class of K 1;3 ­free graphs includes all the line
graphs and is well studied in many aspects. For instance, it is known that, for K 1;3 ­free graphs, the
strong Berge conjecture holds [1], and certain problems that are NP­complete in the general setting
are polynomially solvable for them [2, 3]. Since the middle of the 70s the articles have begun to appear
that are devoted to study of conditions sufficient for hamiltonicity of a K 1;3 ­free graph (see Section 3
of the newest survey by Gould [4]). It is the following conjecture extending Thomassen's conjecture
for line graphs [5] that became widely known.
Conjecture [6]. Every 4­connected K 1;3 ­free graph is Hamiltonian.
It is known that the Hamiltonian cycle problem is NP­complete for 3­connected cubic planar
graphs [7]. Plummer and Pulleyblank [8] showed that there exist infinitely many non­Hamiltonian
3­connected cubic graphs that, moreover, possess no dominating cycles. It is easy to observe that
the replacement of vertices with triangles transforms a 3­connected cubic graph into a 3­connected
K 1;3 ­free graph; moreover, the former is Hamiltonian if and only if such is the latter. Consequently,
the above­cited assertions for 3­connected cubic graphs are also valid for the 3­connected K 1;3 ­free
graphs.

Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

Collections: Mathematics