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- Robert Sámal
- Discrete Mathematics
- 2003

- Robert Sámal
- Electronic Notes in Discrete Mathematics
- 2005

- Matt DeVos, Bojan Mohar, Robert Sámal
- Electronic Notes in Discrete Mathematics
- 2008

The n crossing number of a graph G, denoted crn(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a > b > 0, there exists a graph G for which cr0(G) = a, cr1(G) = b, and cr2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.

- Matt DeVos, Luis A. Goddyn, Bojan Mohar, Robert Sámal
- J. Comb. Theory, Ser. B
- 2009

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “(3,6)-fullerenes”, have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form {λ,−λ} except for the four eigenvalues {3,−1,−1,−1}. We… (More)

- Robert Sámal, Rudolf Stolar, Tomás Valla
- IWOCA
- 2011

The guarding game is a game in which a set of cops has to guard a region in a digraph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of… (More)

- Daniel Král, Jana Maxová, Robert Sámal, Pavel Podbrdský
- Journal of Graph Theory
- 2005

- Matt DeVos, Agelos Georgakopoulos, Bojan Mohar, Robert Sámal
- Discrete & Computational Geometry
- 2010

Eberhard proved that for every sequence (pk), 3 ≤ k ≤ r, k 6= 5, 7 of non-negative integers satisfying Euler’s formula P k≥3(6 − k)pk = 12, there are infinitely many values p6 such that there exists a simple convex polyhedron having precisely pk faces of length k for every k ≥ 3, where pk = 0 if k > r. In this paper we prove a similar statement when… (More)

- Jirí Matousek, Robert Sámal
- Electr. J. Comb.
- 2007

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n ) vertices, where c is a positive constant. The best known upper bound is (2 + o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.

- Zdenek Dvorak, Bojan Mohar, Robert Sámal
- Journal of Graph Theory
- 2013

The star chromatic index χs(G) of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree ∆ = ∆(G). Our best lower bound on χs in terms of ∆ is 2∆(1 + o(1)) valid for complete graphs. We also consider… (More)

- Tomás Kaiser, Daniel Král, Bernard Lidický, Pavel Nejedlý, Robert Sámal
- SIAM J. Discrete Math.
- 2010

The Shortest Cycle Cover Conjecture asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.4m. We show that every bridgeless graph with minimum degree three that contains m edges has a cycle cover comprised of three cycles of total length at most 44m/27 ≈ 1.6296m; this extends a bound of Fan… (More)