4,287 research outputs found

    The independence number in graphs of maximum degree three

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    We prove that a K4-free graph G of order n, size m and maximum degree at most three has an independent set of cardinality at least 1/7 (4n − m − \lambda − tr) where \lambda counts the number of components of G whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and tr is the maximum number of vertex-disjoint triangles in G. Our result generalizes a bound due to Heckman and Thomas (A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237)

    Triangles in graphs without bipartite suspensions

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    Given graphs TT and HH, the generalized Tur\'an number ex(n,T,H)(n,T,H) is the maximum number of copies of TT in an nn-vertex graph with no copies of HH. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex(n,K3,H)(n,K_3,H) when the chromatic number of HH is greater than 3 and proved several results when HH is bipartite. We consider this problem when HH has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension H^\widehat H of a graph HH is the graph obtained from HH by adding a new vertex adjacent to all vertices of HH. We give new upper and lower bounds on ex(n,K3,H^)(n,K_3,\widehat{H}) when HH is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.Comment: New result about path with 5 edges adde

    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

    On two problems in Ramsey-Tur\'an theory

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    Alon, Balogh, Keevash and Sudakov proved that the (k−1)(k-1)-partite Tur\'an graph maximizes the number of distinct rr-edge-colorings with no monochromatic KkK_k for all fixed kk and r=2,3r=2,3, among all nn-vertex graphs. In this paper, we determine this function asymptotically for r=2r=2 among nn-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an nn-vertex KkK_k-free graph GG with α(G)=o(n)\alpha(G)=o(n). The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page
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