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Michael Molloy

Publications and source records attributed to Michael Molloy.

18 recordsLinked to original sources
arXiv2024

Simultaneous edge-colourings

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree $\Delta$, for small values of $k$ and for an infinite sequence of values of $k$. This asymptotically settles a conjecture of Cabello for $k=2$. Moreover, we show that $\sqrt k \Delta + o(\Delta)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell \Delta + o(\Delta)$ colours are enough when every edge appears in at most $\ell$ of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of F\"uredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings.

math.CO↗
arXiv2023

Perfect matchings and loose Hamilton cycles in the semirandom hypergraph model

We study the 2-offer semirandom 3-uniform hypergraph model on $n$ vertices. At each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. We show a strategy that constructs a perfect matching, and another that constructs a loose Hamilton cycle, both succeeding asymptotically almost surely within $\Theta(n)$ steps. Both results extend to $s$-uniform hypergraphs. The challenges with hypergraphs, and our methods, are qualitatively different from what has been seen for semirandom graphs. Much of our analysis is done on an auxiliary graph that is a uniform $k$-out subgraph of a random bipartite graph, and this tool may be useful in other contexts.

math.CO↗
arXiv2022

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).

math.CO↗
arXiv2021

Adaptable and conflict colouring multigraphs with no cycles of length three or four

The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $τ$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring, $σ$, of $G$ such that there is no edge $e = uv$ where $τ(e) = σ(u) = σ(v)$. Here we show that for a multigraph $G$ with maximum degree $Δ$ and no cycles of length 3 or 4, $\mathrm{ch}_a(G) \leq (2\sqrt{2}+o(1))\sqrt{Δ/\lnΔ}$. Under natural restrictions we can show that the same bound holds for the conflict choosability of $G$, which is a closely related parameter defined by Dvořák, Esperet, Kang and Ozeki [arXiv:1803.10962].

math.CO↗
arXiv2019

k-regular subgraphs near the k-core threshold of a random graph

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least $e^{-Θ(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-Θ(k)}$.

math.CO↗
arXiv2019

Fractional cocoloring of graphs

The cochromatic number $Z(G)$ of a graph $G$ is the fewest number of colors needed to color the vertices of $G$ so that each color class is a clique or an independent set. In a fractional cocoloring of $G$ a non-negative weight is assigned to each clique and independent set so that for each vertex $v$, the sum of the weights of all cliques and independent sets containing $v$ is at least one. The smallest total weight of such a fractional cocoloring of $G$ is the fractional cochromatic number $Z_f(G)$. In this paper we prove results for the fractional cochromatic number $Z_f(G)$ that parallel results for $Z(G)$ and the well studied fractional chromatic number $χ_f{(G)}$. For example $Z_f(G)=χ_f(G)$ when $G$ is triangle-free, except when the only nontrivial component of $G$ is a star. More generally, if $G$ contains no $k$-clique, then $Z_f(G)\le χ_f(G)\le Z_f(G)+R(k,k)$. Moreover, every graph $G$ with $χ_f(G)=m$ contains a subgraph $H$ with $Z_f(H)\ge (\frac 14 - o(1))\frac m{\log_2 m}$. We also prove that the maximum value of $Z_f(G)$ over all graphs $G$ of order $n$ is $Θ(n/\log n)$, and the maximum over all graphs embedded on an orientable surface of genus $g$ is $Θ(\sqrt g / \log g)$.

math.CO↗
arXiv2018

The list chromatic number of graphs with small clique number

We prove that every triangle-free graph with maximum degree $Δ$ has list chromatic number at most $(1+o(1))\fracΔ{\ln Δ}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{Δ\ln\lnΔ}{\lnΔ}$.

math.CO↗
arXiv2017

Inside the clustering window for random linear equations

We study a random system of cn linear equations over n variables in GF(2), where each equation contains exactly r variables; this is equivalent to r-XORSAT. Previous work has established a clustering threshold, c^*_r for this model: if c=c_r^*-εfor any constant ε>0 then with high probability all solutions form a well-connected cluster; whereas if c=c^*_r+ε, then with high probability the solutions partition into well-connected, well-separated clusters (with probability tending to 1 as n goes to infinity). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly non-rigorous techniques from statistical physics. We extend that study to the range c=c^*_r+o(1), and prove that the connectivity parameters of the r-XORSAT clusters undergo a smooth transition around the clustering threshold.

cs.CC↗
arXiv2015

The solution space geometry of random linear equations

We consider random systems of linear equations over GF(2) in which every equation binds k variables. We obtain a precise description of the clustering of solutions in such systems. In particular, we prove that with probability that tends to 1 as the number of variables, n, grows: for every pair of solutions σ, τ, either there exists a sequence of solutions σ,...,τ, in which successive elements differ by O(log n) variables, or every sequence of solutions σ,...,τ, contains a step requiring the simultaneous change of Ω(n) variables. Furthermore, we determine precisely which pairs of solutions are in each category. Our results are tight and highly quantitative in nature. Moreover, our proof highlights the role of unique extendability as the driving force behind the success of Low Density Parity Check codes and our techniques also apply to the problem of so-called pseudo-codewords in such codes.

cs.DS↗
arXiv2013

Containing Viral Spread on Sparse Random Graphs: Bounds, Algorithms, and Experiments

Viral spread on large graphs has many real-life applications such as malware propagation in computer networks and rumor (or misinformation) spread in Twitter-like online social networks. Although viral spread on large graphs has been intensively analyzed on classical models such as Susceptible-Infectious-Recovered, there still exits a deficit of effective methods in practice to contain epidemic spread once it passes a critical threshold. Against this backdrop, we explore methods of containing viral spread in large networks with the focus on sparse random networks. The viral containment strategy is to partition a large network into small components and then to ensure the sanity of all messages delivered across different components. With such a defense mechanism in place, an epidemic spread starting from any node is limited to only those nodes belonging to the same component as the initial infection node. We establish both lower and upper bounds on the costs of inspecting inter-component messages. We further propose heuristic-based approaches to partition large input graphs into small components. Finally, we study the performance of our proposed algorithms under different network topologies and different edge weight models.

math.PR↗
arXiv2013

Inside the clustering threshold for random linear equations

We study a random system of $cn$ linear equations over $n$ variables in GF(2), where each equation contains exactly $r$ variables; this is equivalent to $r$-XORSAT. \cite{ikkm,amxor} determined the clustering threshold, $c^*_r$: if $c=c^*_r+\e$ for any constant $\e>0$, then \aas the solutions partition into well-connected, well-separated {\em clusters} (with probability tending to 1 as $n\rightarrow\infty$). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly non-rigorous techniques from statistical physics. We extend that study to the range $c=c^*_r+o(1)$, showing that if $c=c^*_r+n^{-\d}, \d>0$, then the connectivity parameter of each $r$-XORSAT cluster is $n^{Θ(\d)}$, as compared to $O(\log n)$ when $c=c^*_r+\e$. This means that one can move between any two solutions in the same cluster via a sequence of solutions where consecutive solutions differ on at most $n^{Θ(\d)}$ variables; this is tight up to the implicit constant. In contrast, moving to a solution in another cluster requires that some pair of consecutive solutions differ in at least $n^{1-O(\d)}$ variables. Along the way, we prove that in a random $r$-uniform hypergraph with edge-density $n^{-\d}$ above the $k$-core threshold, \aas every vertex not in the $k$-core can be removed by a sequence of $n^{Θ(\d)}$ vertex-deletions in which the deleted vertex has degree less than $k$; again, this is tight up to the implicit constant.

cs.DM↗
arXiv2012

Frozen variables in random boolean constraint satisfaction problems

We determine the exact freezing threshold, r^f, for a family of models of random boolean constraint satisfaction problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large. If the constraint-density of a random CSP, F, in our family is greater than r^f then for almost every solution of F, a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraint-density is less than r^f, then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by non-rigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: that for densities below r^f the random CSPs can be solved using very simple algorithms, while for densities above r^f one requires more sophisticated techniques in order to deal with frozen clusters.

cs.DM↗
arXiv2012

The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k >= 3. The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity.

cs.DM↗
arXiv2009

The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence ${\cal D}$, satisfying certain conditions. We focus on two parameters $Q=Q({\cal D}), R=R({\cal D})$. Molloy and Reed proved that Q=0 is the threshold for the random graph to have a giant component. We prove that if $|Q|=O(n^{-1/3} R^{2/3})$ then, with high probability, the size of the largest component of the random graph will be of order $Θ(n^{2/3}R^{-1/3})$. If $|Q|$ is asymptotically larger than $n^{-1/3}R^{2/3}$ then the size of the largest component is asymptotically smaller or larger than $n^{2/3}R^{-1/3}$. Thus, we establish that the scaling window is $|Q|=O(n^{-1/3} R^{2/3})$.

math.CO↗
arXiv2009

Sharp thresholds for constraint satisfaction problems and homomorphisms

We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction problems with domain size three.

math.CO↗
arXiv2005

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.

cond-mat.stat-mech↗