## Publications

- [Show abstract] [Hide abstract]

**ABSTRACT:**Given a fixed positive integer k ≥ 2, let G be a simple graph of order n ≥ 6k. It is proved that if the minimum degree of G is at least n/2 + 1, then for every pair of vertices x and y, there exists a Hamiltonian cycle such that the distance between x and y along that cycle is precisely k.Graphs and Combinatorics 07/2014; 30(4). DOI:10.1007/s00373-013-1325-9 · 0.39 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We prove that in a random 3-uniform or 4-uniform hypergraph of order n the probability that some two vertices have the same degree tends to one as n→∞.SIAM Journal on Discrete Mathematics 01/2013; 27(1). DOI:10.1137/100785156 · 0.65 Impact Factor -
##### Article: Around a biclique cover conjecture

[Show abstract] [Hide abstract]

**ABSTRACT:**We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every r-coloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can be covered by the vertices of at most 2r-2 monochromatic connected components. We reduce this conjecture to design-like conjectures, where the monochromatic components of the color classes are bicliques [X,Y] with nonempty blocks X and Y. We prove this conjecture for r<6. We show that the width (the number of bicliques) in every color class of any spanning r-coloring is at most 2^{r-1} (and this is best possible). - [Show abstract] [Hide abstract]

**ABSTRACT:**It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The ‘standard’ reasoning is that such a graph must contain a subgraph of minimum degree at least t. Bohman, Frieze and Mubayi noticed that although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Po-Shen Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges.Here we observe that the basic property of the well-known greedy algorithm implies immediately a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r-uniform hypergraph with more than t colors then the hypergraph contains every r-uniform hypertree with t edges.Discrete Mathematics 02/2011; 311(2-3):208-209. DOI:10.1016/j.disc.2010.10.017 · 0.56 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Let f(n;C4) be the smallest integer such that, given any set of edge disjoint quadrilaterals on n vertices, one can extend it into a complete quadrilateral decomposition by including at most f(n;C4) additional vertices. It is known, and it is easy to show, that . Here we settle the longstanding problem that .Journal of Combinatorial Theory Series A 05/2010; 117(4-117):466-474. DOI:10.1016/j.jcta.2009.06.003 · 0.78 Impact Factor -
- [Show abstract] [Hide abstract]

**ABSTRACT:**We show that the four-cycle has a k-fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ⌈5k/3⌉; furthermore, the same is not true with shorter list lengths. In terms of h(k)(G), the k -fold Hall number of a graph G, this result is stated as h(k)(C4)=2k−⌊k/3⌋. For longer cycles it is known that h(k)(Cn)=2k, for n odd, and 2k−⌊k/(n−1)⌋≤h(k)(Cn)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C4). We prove that if G is the diamond (a four-cycle with a diagonal), then h(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.Journal of Graph Theory 12/2009; 65(1):16 - 34. DOI:10.1002/jgt.20462 · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For integers m,k >= 1, we investigate the maximum size of a directed cut in directed graphs in which there are m edges and each vertex has either indegree at most k or outdegree at most k.Journal of Graph Theory 06/2009; 61(2). DOI:10.1002/jgt.20374 · 0.63 Impact Factor -
##### Article: Connected graphs without long paths

[Show abstract] [Hide abstract]

**ABSTRACT:**We determine the maximum number of edges in a connected graph with n vertices if it contains no path with k+1k+1 vertices. We also determine the extremal graphs.Discrete Mathematics 10/2008; 308(19):4487–4494. DOI:10.1016/j.disc.2007.08.047 · 0.56 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r−1)-coloring of the edges of , the complete r-uniform hypergraph on n vertices. We prove the conjecture for r=3,n⩾5 and its asymptotic version for r=4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in ⌊(r−1)/2⌋-colorings of for large n; and a Berge-cycle of order (1−o(1))n in (r−⌊log2r⌋)-colorings of . The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of .Journal of Combinatorial Theory Series B 03/2008; 98(2-98):342-358. DOI:10.1016/j.jctb.2007.07.002 · 0.98 Impact Factor -
- [Show abstract] [Hide abstract]

**ABSTRACT:**We prove that any cycleAKCE International Journal of Graphs and Combinatorics 01/2008; - [Show abstract] [Hide abstract]

**ABSTRACT:**It is easily shown that every digraph with m edges has a directed cut of size at least m/4, and that 1/4 cannot be replaced by any larger constant. We investigate the size of the largest directed cut in acyclic digraphs, and prove a number of related results concerning cuts in digraphs and acyclic digraphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 1–13, 2007Journal of Graph Theory 05/2007; 55(1):1 - 13. DOI:10.1002/jgt.20215 · 0.63 Impact Factor - SIAM Journal on Discrete Mathematics 01/2007; 21(1). DOI:10.1137/S0895480102414107 · 0.65 Impact Factor
- [Show abstract] [Hide abstract]

**ABSTRACT:**For simple graphs G and H, let f(G,H) denote the least integer N such that every coloring of the edges of KN contains either a monochromatic copy of G or a rainbow copy of H. Here we investigate f(G,H) when H = Pk. We show that even if the number of colors is unrestricted when defining f(G,H), the function f(G,Pk), for k = 4 and 5, equals the (k - 2)- coloring diagonal Ramsey number of G. © 2006 Wiley Periodicals, Inc. J Graph TheoryJournal of Graph Theory 01/2007; 54(1):1-12. DOI:10.1002/jgt.20179 · 0.63 Impact Factor -
- [Show abstract] [Hide abstract]

**ABSTRACT:**A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called “on-line coloring.” The properties of on-line colorings are investigated in several classes of graphs. In many cases we find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.Journal of Graph Theory 10/2006; 12(2):217 - 227. DOI:10.1002/jgt.3190120212 · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**It has been communicated by P. Manca in this journal that all 4-regular connected planar graphs can be generated from the graph of the octahedron using simple planar graph operations. We point out an error in the generating procedure and correct it by including an additional operation.Journal of Graph Theory 10/2006; 5(4):423 - 426. DOI:10.1002/jgt.3190050412 · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A Θ-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H=(V,E) has no Θ-cycle, and |e|⩾3, for every edge e∈E, then ∑e∈E(|e|-1)⩽2|V|-2 with equality if and only if H is obtained from a hypertree by doubling its edges.This result reminiscent of Berge's and Lovász's similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have Θ-cycles, and 3-uniform simple hypergraphs with n vertices and n-1 edges have Θ-cycles. Both results are sharp. Since the presence of a Θ-cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for Θ-cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have Θ-cycles when |E|⩾5n/6-c1n and have odd cycles when |E|⩾7n/9-c2n and these are sharp results apart from the values of the constants.Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results—in disguise—are extremal results for bipartite graphs that have no cycles with chords.Discrete Mathematics 10/2006; 306(19):2481-2491. DOI:10.1016/j.disc.2005.12.037 · 0.56 Impact Factor -
##### Article: Hall ratio of the Mycielski graphs

[Show abstract] [Hide abstract]

**ABSTRACT:**Let n (G) denote the number of vertices of a graph G and let a (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by [GRAPHICS] where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies omega(G) <= rho(G) <= chi(G) where omega and chi denote the clique number and the chromatic number of G, respectively. We show that the interval [omega(G), rho(G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs. (c) 2006 Elsevier B.V. All rights reserved.Discrete Mathematics 08/2006; 306(16):1988-1990. DOI:10.1016/j.disc.2005.09.020 · 0.56 Impact Factor

- Questions & Answers
- Open Reviews
- Research Feedback