196 research outputs found

    Planar Ramsey numbers for cycles

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    AbstractFor two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles

    Improved FPT algorithms for weighted independent set in bull-free graphs

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    Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2O(k5)⋅n92^{O(k^5)} \cdot n^9. In this article we improve this running time to 2O(k2)⋅n72^{O(k^2)} \cdot n^7. As a byproduct, we also improve the previous Turing-kernel for this problem from O(k5)O(k^5) to O(k2)O(k^2). Furthermore, for the subclass of bull-free graphs without holes of length at most 2p−12p-1 for p≥3p \geq 3, we speed up the running time to 2O(k⋅k1p−1)⋅n72^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7. As pp grows, this running time is asymptotically tight in terms of kk, since we prove that for each integer p≥3p \geq 3, Weighted Independent Set cannot be solved in time 2o(k)⋅nO(1)2^{o(k)} \cdot n^{O(1)} in the class of {bull,C4,…,C2p−1}\{bull,C_4,\ldots,C_{2p-1}\}-free graphs unless the ETH fails.Comment: 15 page

    On the Geometric Thickness of 2-Degenerate Graphs

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    A graph is 2-degenerate if every subgraph contains a vertex of degree at most 2. We show that every 2-degenerate graph can be drawn with straight lines such that the drawing decomposes into 4 plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of 2-degenerate graphs is at most 4. On the other hand, we show that there are 2-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into 2 plane graphs. That is, there are 2-degenerate graphs with geometric thickness, and hence geometric arboricity, at least 3. This answers two questions posed by Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004]

    Graph Arrowing: Constructions and Complexity

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    Graph arrowing is concerned with determining which monochromatic subgraphs are unavoidable when coloring a given graph. There are two main avenues of research concerning arrowing: finding extremal Ramsey/Folkman graphs and categorizing the complexity of arrowing problems. Both avenues have been studied extensively for decades. In this thesis, we focus on graph arrowing problems where one of the monochromatic subgraphs being avoided is the path on three vertices, denoted as P3. Our main contributions involve computing Folkman numbers by generating graphs up to 13 vertices and proving the coNP-completeness of some arrowing problems using a novel reduction framework geared towards avoiding P3\u27s. The (P3,H)-Arrowing Problem asks whether a given graph can be colored using two colors (red and blue) such that there are no red P3\u27s and no blue H\u27s, where H is a fixed graph. The few previous hardness proofs for arrowing problems relied on ad-hoc, laborious constructions of gadgets. We introduce a general framework that can be used to prove the coNP-completeness of (P3,H)-arrowing problems. We search for gadgets computationally. These gadgets allow us to simulate variants of SAT, thus showing coNP-hardness. Finally, we use our (P3,H)-Arrowing hardness reductions to gain insight into variants of Monotone SAT. For fixed k in {4,5,6}, we show that Monotone SAT remains NP-complete under the following constraints: 1) each clause consists of exactly two unnegated literals or exactly k negated literals, 2) the variables in each clause are distinct, and 3) the number of times a variable occurs in the formula is bounded by a constant. For future work, we expect that the insight gained by our computationally assisted reductions will help us prove the complexity of other elusive arrowing problems
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