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We abstract and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph G=(V,E) and a set of demand pairs P⊆VxV, a reachability preserver is a sparse subgraph H that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an n-node graph and demand pairs of the form P⊆SxV for a small node subset S, there is always a reachability preserver on O(n+√n/P//S/) edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which O(n) size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0.6+ε) of Chlamatac, et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534-443] (n4/7+ε). [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least 2k 1 contains k vertex-disjoint cycles. This conjecture is famous as one of a hundred unsolved problems selected in [A. Bondy and M. R. Murty, Graph Theory, Springer-Verlag, London, 2008]. Lichiardopol, P\'or, and Sereni proved in [SIAM J. Discrete Math., 23 (2009), pp. 979--992] that the above conjecture holds for k = 3. Let g be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy, and Thomass\'e conjectured in [J. Graph Theory, 75 (2014), pp. 284--302] that every digraph with girth g and minimum outdegree at least g g 1 k contains k vertex-disjoint cycles. Thomass\'e conjectured around 2006 that every oriented graph (a digraph without 2-cycles) with girth g and minimum outdegree at least h contains a path of length h(g 1), where h is a positive integer. In this paper, we first present a new shorter proof of the Bermond--Thomassen conjecture for the case of k = 3, and then we disprove the conjecture proposed by Bang-Jensen, Bessy, and Thomass\'e. Finally, we disprove the even girth case of the conjecture proposed by Thomass\'e. [ABSTRACT FROM AUTHOR]

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg [SIAM J. Discrete Math., 5 (1992), pp. 398--412] states that the queue number of planar graphs is bounded. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has bounded track number, and that every k-planar graph (i.e., every graph that can be drawn in the plane with at most k crossings per edge) of bounded degree has bounded queue number. [ABSTRACT FROM AUTHOR]

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak and Tuza [SIAM J. Discrete Math., 7 (1994), pp. 307--313] and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon, Krivelevich, and Sudakov [J. Combin. Theory Ser. B, 77 (1999), pp. 73--82] bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for X (G) as a function of the number of vertices n, the number of edges m, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds. [ABSTRACT FROM AUTHOR]

We consider the symmetric difference of two graphs on the same vertex set [n], which is the graph on [n] whose edge set consists of all edges that belong to exactly one of the two graphs. Let F be a class of graphs, and let MF(n) denote the maximum possible cardinality of a family G of graphs on [n] such that the symmetric difference of any two members in G belongs to F. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379-403] with the aim of providing a new graphic approach to coding theory. In particular, MF(n) denotes the maximum possible size of this code. Existing results show that as the graph class F changes, MF(n) can vary from n to 2(1+0(1))(2n). We study several phase transition problems related to MF(n) in general settings and present a partial solution to a recent problem posed by Alon et al. [ABSTRACT FROM AUTHOR]

For integers k ≥1 and m ≥2, let g(k,m) be the least integer n ≥1 such that every graph with chromatic number at least n contains a (k + 1)-connected subgraph with chromatic number at least m. Refining the recent result of Gir\~ao and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868--875] that g(k 1, k) ≤ 7k + 1 for all k≥ 2, we prove that g(k,m) ≤ max(m + 2k 2, (3 + 1/16)k≤) for all k ≥ 1 and m≥2. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367-371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567-586], and of Penev, Thomass'e, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592--619]. Our result implies that g(k, k + 1) (3 + 1/16)k for all k \geq 1, making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261--271] that g(k, k +1)≤3k +1, which was originally a result with a false proof and was the starting point of this research area. [ABSTRACT FROM AUTHOR]

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al. [Maximally recoverable codes for grid-like topologies, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 2092--2108] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph Kn,n, with labels coming from F2d, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n×n grid topologies. Prior to the current work, it was known that d is between (log n)² and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group. [ABSTRACT FROM AUTHOR]