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Minimal unavoidable sets of cycles in plane graphs

Madaras T., Tamasova M.

Opuscula Mathematica

2018

Vol. 38, no. 6

859--870

A set S of cycles is minimal unavoidable in a graph family [formula] if each graph [formula] contains a cycle from S and, for each proper subset S' ⊂ S, there exists an infinite subfamily [formula] such that no graph from [formula] contains a cycle from S'. In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles

Forbidden configurations for hypohamiltonian graphs

Fabrici I., Madaras T., Timkova M.

Opuscula Mathematica

2018

Vol. 38, no. 3

357--377

A graph G is called hypohamiltonian if G is not hamiltonian, but G — x is hamiltonian for each vertex x of G. We present a list of 331 forbidden configurations which do not appear in hypohamiltonian graphs.

On the dimension of Archimedean solids

Madaras T., Siroczki P.

Opuscula Mathematica

2014

Vol. 34, no. 1

123--138

We study the dimension of graphs of the Archimedean solids. For most of these graphs we find the exact value of their dimension by finding unit-distance embeddings in the euclidean plane or by proving that such an embedding is not possible.

Notes on topological indices of graph and its complement

Madaras T., Mockovciakova M.

Opuscula Mathematica

2013

Vol. 33, no. 1

107-115

In this note, we derive the lower bound on the sum for Wiener index of bipartite graph and its bipartite complement, as well as the lower and upper bounds on this sum for the Randić index and Zagreb indices. We also discuss the quality of these bounds.