46 research outputs found

    Hardness Transitions of Star Colouring and Restricted Star Colouring

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    We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For k∈Nk\in \mathbb{N}, a kk-colouring of a graph GG is a function f ⁣:V(G)β†’Zkf\colon V(G)\to \mathbb{Z}_k such that f(u)β‰ f(v)f(u)\neq f(v) for every edge uvuv of GG. A kk-colouring of GG is called a kk-star colouring of GG if there is no path u,v,w,xu,v,w,x in GG with f(u)=f(w)f(u)=f(w) and f(v)=f(x)f(v)=f(x). A kk-colouring of GG is called a kk-rs colouring of GG if there is no path u,v,wu,v,w in GG with f(v)>f(u)=f(w)f(v)>f(u)=f(w). For k∈Nk\in \mathbb{N}, the problem kk-STAR COLOURABILITY takes a graph GG as input and asks whether GG admits a kk-star colouring. The problem kk-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of kk-star colouring and kk-rs colouring with respect to the maximum degree for all kβ‰₯3k\geq 3. For kβ‰₯3k\geq 3, let us denote the least integer dd such that kk-STAR COLOURABILITY (resp. kk-RS COLOURABILITY) is NP-complete for graphs of maximum degree dd by Ls(k)L_s^{(k)} (resp. Lrs(k)L_{rs}^{(k)}). We prove that for k=5k=5 and kβ‰₯7k\geq 7, kk-STAR COLOURABILITY is NP-complete for graphs of maximum degree kβˆ’1k-1. We also show that 44-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and kk-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree kβˆ’1k-1 for kβ‰₯5k\geq 5. Using these results, we prove the following: (i) for kβ‰₯4k\geq 4 and d≀kβˆ’1d\leq k-1, kk-STAR COLOURABILITY is NP-complete for dd-regular graphs if and only if dβ‰₯Ls(k)d\geq L_s^{(k)}; and (ii) for kβ‰₯4k\geq 4, kk-RS COLOURABILITY is NP-complete for dd-regular graphs if and only if Lrs(k)≀d≀kβˆ’1L_{rs}^{(k)}\leq d\leq k-1

    Extending partial edge colorings of iterated cartesian products of cycles and paths

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    We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if G=C2kdG=C^d_{2k} is the ddth power of the cartesian product of the even cycle C2kC_{2k} with itself, and at most 2dβˆ’12d-1 edges of GG are precolored, then there is a proper 2d2d-edge coloring of GG that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least 44 from each other. For odd cycles of length at least 55, we prove that if G=C2k+1dG=C^d_{2k+1} is the ddth power of the cartesian product of the odd cycle C2k+1C_{2k+1} with itself (kβ‰₯2k\geq2), and at most 2d2d edges of GG are precolored, then there is a proper (2d+1)(2d+1)-edge coloring of GG that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444]

    Packing Arc-Disjoint 4-Cycles in Oriented Graphs

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    Given a directed graph G and a positive integer k, the Arc Disjoint r-Cycle Packing problem asks whether G has k arc-disjoint r-cycles. We show that, for each integer r ? 3, Arc Disjoint r-Cycle Packing is NP-complete on oriented graphs with girth r. When r is even, the same result holds even when the input class is further restricted to be bipartite. On the positive side, focusing on r = 4 in oriented graphs, we study the complexity of the problem with respect to two parameterizations: solution size and vertex cover size. For the former, we give a cubic kernel with quadratic number of vertices. This is smaller than the compression size guaranteed by a reduction to the well-known 4-Set Packing. For the latter, we show fixed-parameter tractability using an unapparent integer linear programming formulation of an equivalent problem

    Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

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    In 1973, Erd\H{o}s conjectured the existence of high girth (n,3,2)(n,3,2)-Steiner systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erd\H{o}s' conjecture. As for Steiner systems with more general parameters, Glock, K\"{u}hn, Lo, and Osthus conjectured the existence of high girth (n,q,r)(n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph GG avoiding a given set of submatchings (which we view as a hypergraph HH where V(H)=E(G)V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Koml\'os, Pintz, Spencer, and Szemer\'edi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when HH is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of KnrK_n^r into approximate high girth (n,q,r)(n,q,r)-Steiner systems. Our main results also imply the existence of a perfect matching in a bipartite hypergraph where the parts have slightly unbalanced degrees. This has a number of applications; for example, it proves the existence of Ξ”\Delta pairwise disjoint list colorings in the setting of Kahn's theorem; it also proves asymptotic versions of various rainbow matching results in the sparse setting (where the number of times a color appears could be much smaller than the number of colors) and even the existence of many pairwise disjoint rainbow matchings in such circumstances.Comment: 52 page

    The degree-restricted random process is far from uniform

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    The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph. In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before).Comment: 32 pages, 3 figure

    Partially Oriented 6-star Decomposition of Some Complete Mixed Graphs

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    Let MvM_v denotes a complete mixed graph on vv vertices, and let S6iS_6^i denotes the partial orientation of the 6-star with twice as many arcs as edges. In this work, we state and prove the necessary and sufficient conditions for the existence of λ\lambda-fold decomposition of a complete mixed graph into S6iS_6^i for i∈{1,2,3,4}i\in\{1,2,3,4\}. We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of S6iS_6^i-decomposition of the complete bipartite mixed graph for i∈{1,2,3,4}i\in\{1,2,3,4\}. Finally, this work introduces the decomposition of a complete mixed graph with a hole into mixed stars

    Almost-Orthogonal Bases for Inner Product Polynomials

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    In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products. We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative

    Cut Sparsification of the Clique Beyond the Ramanujan Bound: A Separation of Cut Versus Spectral Sparsification

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    We prove that a random dd-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most (22Ο€+on,d(1))/d\left(2\sqrt{\frac 2 \pi} + o_{n,d}(1)\right)/\sqrt d, where 22Ο€=1.595…2\sqrt{\frac 2 \pi} = 1.595\ldots and on,d(1)o_{n,d}(1) denotes an error term that depends on nn and dd and goes to zero if we first take the limit nβ†’βˆžn\rightarrow \infty and then the limit dβ†’βˆžd \rightarrow \infty. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prove new lower bounds on spectral sparsification of the clique. If GG is a spectral sparsifier of the clique and GG has average degree dd, we prove that the approximation error is at least the "Ramanujan bound'' (2βˆ’on,d(1))/d(2-o_{n,d}(1))/\sqrt d, which is met by dd-regular Ramanujan graphs, provided that either the weighted adjacency matrix of GG is a (multiple of) a doubly stochastic matrix, or that GG satisfies a certain high "odd pseudo-girth" property. The first case can be seen as an "Alon-Boppana theorem for symmetric doubly stochastic matrices," showing that a symmetric doubly stochastic matrix with dndn non-zero entries has a non-trivial eigenvalue of magnitude at least (2βˆ’on,d(1))/d(2-o_{n,d}(1))/\sqrt d; the second case generalizes a lower bound of Srivastava and Trevisan, which requires a large girth assumption. Together, these results imply a separation between spectral sparsification and cut sparsification. If GG is a random log⁑n\log n-regular graph on nn vertices, we show that, with high probability, GG admits a (weighted subgraph) cut sparsifier of average degree dd and approximation error at most (1.595…+on,d(1))/d(1.595\ldots + o_{n,d}(1))/\sqrt d, while every (weighted subgraph) spectral sparsifier of GG having average degree dd has approximation error at least (2βˆ’on,d(1))/d(2-o_{n,d}(1))/\sqrt d.Comment: To appear in SODA 202
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