46 research outputs found
Hardness Transitions of Star Colouring and Restricted Star Colouring
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 , a -colouring of a graph is a function
such that for every edge of
. A -colouring of is called a -star colouring of if there is
no path in with and . A -colouring of
is called a -rs colouring of if there is no path in with
. For , the problem -STAR COLOURABILITY
takes a graph as input and asks whether admits a -star colouring.
The problem -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 -star colouring
and -rs colouring with respect to the maximum degree for all . For
, let us denote the least integer such that -STAR COLOURABILITY
(resp. -RS COLOURABILITY) is NP-complete for graphs of maximum degree by
(resp. ).
We prove that for and , -STAR COLOURABILITY is NP-complete
for graphs of maximum degree . We also show that -RS COLOURABILITY is
NP-complete for planar 3-regular graphs of girth 5 and -RS COLOURABILITY is
NP-complete for triangle-free graphs of maximum degree for .
Using these results, we prove the following: (i) for and ,
-STAR COLOURABILITY is NP-complete for -regular graphs if and only if
; and (ii) for , -RS COLOURABILITY is NP-complete
for -regular graphs if and only if
Extending partial edge colorings of iterated cartesian products of cycles and paths
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 is the th power of the cartesian
product of the even cycle with itself, and at most edges of
are precolored, then there is a proper -edge coloring of 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 from each other. For odd cycles of length at least
, we prove that if is the th power of the cartesian
product of the odd cycle with itself (), and at most
edges of are precolored, then there is a proper -edge coloring of
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
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
In 1973, Erd\H{o}s conjectured the existence of high girth -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 -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 avoiding a given set of
submatchings (which we view as a hypergraph where ). 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 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 into approximate high
girth -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
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
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
Let denotes a complete mixed graph on vertices, and let 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 -fold decomposition of a complete mixed graph into for . We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of -decomposition of the complete bipartite mixed graph for . Finally, this work introduces the decomposition of a complete mixed graph with a hole into mixed stars
Almost-Orthogonal Bases for Inner Product Polynomials
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
We prove that a random -regular graph, with high probability, is a cut
sparsifier of the clique with approximation error at most , where
and denotes an error term that depends on and and goes to
zero if we first take the limit and then the limit .
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
is a spectral sparsifier of the clique and has average degree , we
prove that the approximation error is at least the "Ramanujan bound''
, which is met by -regular Ramanujan graphs,
provided that either the weighted adjacency matrix of is a (multiple of) a
doubly stochastic matrix, or that 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 non-zero entries has a non-trivial eigenvalue of
magnitude at least ; 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 is a random -regular graph on
vertices, we show that, with high probability, admits a (weighted subgraph)
cut sparsifier of average degree and approximation error at most
, while every (weighted subgraph) spectral
sparsifier of having average degree has approximation error at least
.Comment: To appear in SODA 202