Robert Sámal

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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.
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)
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)
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)
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)