Vitaly I Voloshin | Troy University (original) (raw)

Papers by Vitaly I Voloshin

SIAM Journal on Discrete Mathematics, 1998

Recently in several papers, graphs with maximum neighborhood orderings were characterized and tur... more Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified.

Discrete Applied Mathematics, 1996

It is well-known that the incidence graphs of totally balanced hypergraphs are exactly chordal bi... more It is well-known that the incidence graphs of totally balanced hypergraphs are exactly chordal bipartite graphs. This paper examines the incidence graphs of biacyclic hypergraphs. We characterize these graphs as absolute bipartite retracts with forbidden isometric wheels, or alternatively via an elimination scheme.

Bolyai Society Mathematical Studies

We survey results and open problems on 'mixed hypergraphs' that are hypergraphs with two types of... more We survey results and open problems on 'mixed hypergraphs' that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must not be completely multicolored. Though the first condition just means 'classical' hypergraph coloring, its combination with the second one causes rather unusual behavior. For instance, hypergraphs occur that are uncolorable, or that admit colorings with certain numbers k and k of colors but no colorings with exactly k colors for any k < k < k .

In [Discrete Math. 174, (1997) 247-259] an infinite class of STSs(2 h − 1) was found with the upp... more In [Discrete Math. 174, (1997) 247-259] an infinite class of STSs(2 h − 1) was found with the upper chromatic numberχ = h. We prove that in this class, for all STSs(2 h − 1) with h < 10, the lower chromatic number coincides with the upper chromatic number, i.e. χ =χ = h; and moreover, there exists a infinite sub-class of STSs with χ =χ = h for any value of h.

The paper surveys problems, results and methods concerning the coloring of Steiner triple and qua... more The paper surveys problems, results and methods concerning the coloring of Steiner triple and quadruple systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more, restrictively, (ii) a triple of vertices that meets precisely two color classes.

We introduce the notion of a co-edge of a hypergraph, which is a subset of vertices to be colored... more We introduce the notion of a co-edge of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both edges and co-edges are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by ¯ χ(H). An algorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the color-ings of some classes of hypergraphs are discussed. A greedy polynomial time algorithm for finding a lower bound for ¯ χ(H) of a hypergraph H containing only co-edges is presented. The cardinality of a maximum stable set of an all-vertex partial hy-pergraph generated by co-edges is called the co-stability number α A (H). A hypergraph H is called co-perfect if ¯ χ(H) = α A (H) for all its wholly-edge subhypergraphs H. Two classes of minimal non co-perfect hyper-graphs (the so called monostars and cycloids C r 2r−1 , r ≥ 3) are found. It is proved that hypertrees are co-perfect if and only if they do not contain monostars as wholly-edge subhypergraphs. It is conjectured that the r−uniform hypergraph H is co-perfect if and only if it contains neither monostars nor cycloids C r 2r−1 , r ≥ 3, as wholly-edge subhypergraphs.

In this note we consider a finite graph without loops and multiple edges. The colouring of a grap... more In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

We consider the colorings of the edges of a multigraph in such a way that every non-pendant verte... more We consider the colorings of the edges of a multigraph in such a way that every non-pendant vertex is incident to at least two edges of the same color. The maximum number of colors that can be used in such colorings is the upper chromatic index of a multigraph G, denoted byχ (G). We prove that if a multigraph G has n vertices, m edges, p pendant vertices and maximum number c disjoint cycles, thenχ (G) = c + m − n + p.

Dedicated to 90th anniversary of Alexander Zykov

SIAM Journal on Discrete Mathematics, 2006

A Note on Quasi‐triangulated Graphs. [SIAM Journal on Discrete Mathematics 20, 597 (2006)]. Ion G... more A Note on Quasi‐triangulated Graphs. [SIAM Journal on Discrete Mathematics 20, 597 (2006)]. Ion Gorgos, Chính T. Hoàng, Vitaly Voloshin. Abstract. A graph is quasi‐triangulated if each of its induced subgraphs has a vertex ...

Rendiconti Del Seminario Matematico Di Messina, 2003

Discrete Applied Mathematics, Jun 1, 1997