Improved Ramsey-type results for comparability graphs

Korándi, Dániel; Tomon, István

doi:10.48550/arxiv.1810.00588

Cited by 2 publications

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References 16 publications

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“…However, this alone will not be sufficient to find a linear-sized homogeneous submatrix. Indeed, as was shown recently by Korándi and Tomon[21], the size of the bipartite graph in Theorem 6.1 cannot be replaced by anything larger than Ω(n/(log n) k ).However, one can find slightly larger all-0 submatrices in Lemma 6.2 by reducing the number of partial orders we use. For example, we may assume that K in the proof has size at most k − 1, as…”

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confidence: 77%

Large homogeneous submatrices

Korándi¹,

Pach²,

Tomon³

2019

Preprint

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A matrix is homogeneous if all of its entries are equal. Let P be a 2×2 zero-one matrix that is not homogeneous. We prove that if an n×n zero-one matrix A does not contain P as a submatrix, then A has an cn × cn homogeneous submatrix for a suitable constant c > 0. We further provide an almost complete characterization of the matrices P (missing only finitely many cases) such that forbidding P in A guarantees an n 1−o(1) × n 1−o(1) homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane-arrangements and string graphs.

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confidence: 77%

Large homogeneous submatrices

Korándi¹,

Pach²,

Tomon³

2019

Preprint

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“…The rest of this section is devoted to the proof of this theorem. The proof is probabilistic and is inspired by a construction of Korándi and Tomon [20]. We shall consider a random double-ordered…”

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confidence: 99%

On the chromatic number of disjointness graphs of curves

Pach

Tomon

2020

Journal of Combinatorial Theory, Series B

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Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G) = k, then χ(G) ≤ k+12 . If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G) ≤ k+1 2 k+2 3 . Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist K k -free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k 4 ) colors. This matches the upper bound up to a constant factor. function f (k) = 4k 2 . (It is conjectured that the same is true with bounding function f (k) = O(k).) However, an ingenious construction of Burling [4] shows that the family of intersection graphs of axis-parallel boxes in R 3 is not χ-bounded. The χ-boundedness of intersection graphs of chords of a circle was established by Gyárfás [16,17]; see also Kostochka et al. [21,23]. Deciding whether a family of graphs is χ-bounded is often a difficult task [22].Computing the chromatic number of the disjointness graph of a family of objects, C, is equivalent to determining the clique cover number of the corresponding intersection graph G, that is, the minimum number of cliques whose vertices together cover the vertex set of G. This problem can be solved in polynomial time only for some very special families (for instance, if C consists of intervals along a line or arcs along a circle [15]). On the other hand, the problem is known to be NP-complete if C is a family of chords of a circle [18,14] or a family of unit disks in the plane [43,6], and in many other cases. There is a vast literature providing approximation algorithms or inapproximability results for the clique cover number [8,9]. Families of curves.A curve or string in R 2 is the image of a continuous function φ : [0, 1] → R d . A curve C ⊂ R 2 is called x-monotone if every vertical line intersects C in at most one point. We say that C is grounded at the curve L if one of the endpoints of C is in L, and this is the only intersection point of C and L. A grounded x-monotone curve is an x-monotone curve that is contained in the half-plane {x ≥ 0}, and whose left endpoint lies on the vertical line {x = 0}.It was first suggested by Erdős in the 1970s, and remained the prevailing conjecture for 40 years, that the family of intersection graphs of curves (the family of so-called "string graphs") is χ-bounded [3,24]. There were many promising facts pointing in this direction. Extending earlier results of McGuinness [31], Suk [42], and Lasoń ...

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Improved Ramsey-type results for comparability graphs

Cited by 2 publications

References 16 publications

Large homogeneous submatrices

Large homogeneous submatrices

On the chromatic number of disjointness graphs of curves

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