495 research outputs found

    Transversal numbers of uniform hypergraphs

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    Spectrum of mixed bi-uniform hypergraphs

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    A mixed hypergraph is a triple H=(V,C,D)H=(V,\mathcal{C},\mathcal{D}), where VV is a set of vertices, C\mathcal{C} and D\mathcal{D} are sets of hyperedges. A vertex-coloring of HH is proper if CC-edges are not totally multicolored and DD-edges are not monochromatic. The feasible set S(H)S(H) of HH is the set of all integers, ss, such that HH has a proper coloring with ss colors. Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a characterization of feasible sets for mixed hypergraphs with all CC- and DD-edges of the same size rr, r≥3r\geq 3. In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all CC-edges of size ℓ\ell and all DD-edges of size mm, where ℓ,m≥2\ell, m \geq 2. Moreover, we show that for every sequence (r(s))s=ℓn(r(s))_{s=\ell}^n, n≥ℓn \geq \ell, of natural numbers there exists such a hypergraph with exactly r(s)r(s) proper colorings using ss colors, s=ℓ,…,ns = \ell,\ldots,n, and no proper coloring with more than nn colors. Choosing ℓ=m=r\ell = m=r this answers a question of Bujt\'as and Tuza, and generalizes their result with a shorter proof.Comment: 9 pages, 5 figure

    On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed

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    For r≥2r\geq2, let (P,L)(P,\mathcal{L}) be an rr-uniform linear system. The transversal number τ(P,L)\tau(P,\mathcal{L}) of (P,L)(P,\mathcal{L}) is the minimum number of points that intersect every line of (P,L)(P,\mathcal{L}). The 2-packing number ν2(P,L)\nu_2(P,\mathcal{L}) of (P,L)(P,\mathcal{L}) is the maximum number of lines such that the intersection of any three of them is empty. In [Discrete Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it true that if (P,L)(P,\mathcal{L}) is a rr-uniform linear system then τ(P,L)≤∣P∣+∣L∣r+1\tau(P,\mathcal{L})\leq\displaystyle\frac{|P|+|\mathcal{L}|}{r+1} holds for all k≥2k\geq2?. In this paper, some results about of rr-uniform linear systems whose 2-packing number is fixed which satisfies the inequality are given

    Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

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    Given a simple graph G=(V,E)G=(V,E), a subset of EE is called a triangle cover if it intersects each triangle of GG. Let νt(G)\nu_t(G) and τt(G)\tau_t(G) denote the maximum number of pairwise edge-disjoint triangles in GG and the minimum cardinality of a triangle cover of GG, respectively. Tuza conjectured in 1981 that τt(G)/νt(G)≤2\tau_t(G)/\nu_t(G)\le2 holds for every graph GG. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set TG\mathscr T_G of triangles covers all edges in GG. We show that a triangle cover of GG with cardinality at most 2νt(G)2\nu_t(G) can be found in polynomial time if one of the following conditions is satisfied: (i) νt(G)/∣TG∣≥13\nu_t(G)/|\mathscr T_G|\ge\frac13, (ii) νt(G)/∣E∣≥14\nu_t(G)/|E|\ge\frac14, (iii) ∣E∣/∣TG∣≥2|E|/|\mathscr T_G|\ge2. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm
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