Consider the following total order: order the vertices by repeatedly removing a vertex of minimum degree in the subgraph of vertices not yet chosen and placing it after all the remaining vertices but before all the vertices already removed. For which graphs the greedy algorithm on this order gives an optimum vertex-coloring? Markossian, Gasparian and Reed introduced the class of -perfect graphs...

By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many nontrivial complete Hamiltonian minimal submanifolds in CP n and Cn.

We describe and analyze a construction of a vertex cover (consisting of subcubes) in a random subgraph of the n-cube. The main idea of the construction is to select subcubes with minimal intersection into the vertex cover. We estimate the upper bound of such a vertex cover. Our analysis gives a theoretical justification for a heuristic that minimizes the disjunctive normal form of a random Bool...

A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size χ(G) − 1. It is shown that...

OF THE DISSERTATION On Generation of Cut Conjunctions, Minimal k-Connected Spanning Subgraphs, Minimal Connected and Spanning Subsets and Vertices by Konrad Borys Dissertation Director: Professor Endre Boros We consider the following problems: • Cut conjunctions in graphs: given an undirected graphG = (V,E) and a collection of vertex pairs B ⊆ V × V generate all minimal edge sets X ⊆ E such tha...

Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence Λ1, Λ2, . . . , Λn of graph of groups decompositions of G with edge groups in C such that Λ1 is the trivial decomposition (with 1-vertex) and for i > 1, Λi is obtained from Λi−1 by non-trivially and compatibly splitting a vertex group of Λi−1 over a group in C, replacing t...

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and ...

The cyclic antibandwidth problem is to embed an n-vertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a variant of obnoxious facility location problems or a dual problem to the cyclic bandwidth problem. The problem is NP-hard. In the paper we start investigating this invariant for typical graphs. We prove basic facts ...

A subset Q ⊆ V (G) is a dominating set of a graph G if each vertex in V (G) is either in Q or is adjacent to a vertex in Q. A dominating set Q of G is minimal if Q contains no dominating set of G as a proper subset. In this paper we study the number of minimal dominating sets in some classes of trees. Mathematics Subject Classification: 05C69

The application of IR 786 perchlorate (IR-786) as a selective optical sensor for cyanide anion in both organic solution (acetonitrile (MeCN), 100%) and solvent-free solid surfaces was demonstrated. In MeCN, IR-786 was selective to two anions in the following order: CN(-) > OH(-). A significant change in the characteristic dark green color of IR-786 in MeCN to yellow was observed as a result of ...