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Nicholas Crawford

Publications and source records attributed to Nicholas Crawford.

At least 19 recordsLinked to original sources
arXiv2025

On Circuit Imbalance and 0/1 Circuits for Coloring and Spanning Forest Problems

Circuits are fundamental objects in linear programming and oriented matroid theory, representing the elementary difference vectors of a polyhedron between points in its affine space. A recent concept introduced by Ekbatani, Natura, and V\'egh, the circuit imbalance, serves as a complexity measure relevant to iteration bounds for circuit-based augmentation and circuit diameters, as well as the general interpretability of circuits in terms of the underlying application. In this paper, we analyze linear programming formulations of relaxed combinatorial optimization problems to prove two contrasting types of results related to the circuit imbalance. On one hand, we identify simple and common constraint structures, in particular arising in graph-theoretic problems, that inherently lead to an exponential circuit imbalance. These constructions show that, in quite general situations, working with the entire set of circuits poses significant challenges for an application of circuit augmentation or the study of circuit diameters. On the other hand, through a case study of two classic graph-theoretic problems with exponential imbalance, the vertex graph coloring problem and the maximum weight forest problem, we exhibit the existence of sets and subsets of highly interpretable circuits of (best-case) imbalance 1. These sets correspond to the recoloring of vertices or to the addition or removal of edges, respectively, for example generalizing classic concepts of Kempe dynamics in coloring. Their interpretability in terms of the underlying application facilitates a study of circuit walks in the corresponding polytopes. We prove that a restriction of circuit walks to these sets suffices to not only guarantee reachability of the integral extreme-points of the skeleton, but leads to linear and constant circuit diameter bounds, respectively.

math.OC↗
arXiv2025

Cop number of partial cubes

The game of Cops and Robbers on graphs is a well-studied pursuit--evasion model whose central parameter, the cop number, captures the minimum number of pursuers required to guarantee capture of an adversary on a given graph. While the cop number has been determined for many classical graph families, relatively little is known about the important class of partial cubes, i.e., isometric subgraphs of hypercubes. In this paper, we establish a lower bound for the cop number of partial cubes and present an upper bound on a subclass of partial cubes. Additionally, we improve these bounds for a particular family of partial cubes: Fibonacci cubes. These graphs are defined as induced subgraphs of hypercubes obtained by forbidding consecutive ones in binary strings. Beyond their natural combinatorial interest, Fibonacci cubes have connections to chemical graph theory, where they serve as models for resonance graphs of certain classes of polycyclic aromatic hydrocarbons.

math.CO↗
arXiv2025

Rainbow Trees in Hypercubes

We prove that every proper edge-coloring of the $n$-dimensional hypercube $Q_n$ contains a rainbow copy of every tree $T$ on at most $n$ edges. This result is best possible, as $Q_n$ can be properly edge-colored using only $n$ colors while avoiding rainbow cycles.

math.CO↗
arXiv2025

Odd Ramsey numbers of multipartite graphs and hypergraphs

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges. Generalizing a result of \cite{BHZ} in two different ways, in this paper we prove $r_{odd} \left(K_{n,n}, K_{2,t} \right)=\frac{n}{t} + o(n)$ for all $t\geq 2$, and $r_{odd} \left(\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2} \right) = \frac{n}{2} + o(n)$ for all $k\geq 2$. The latter is the first result studying odd Ramsey numbers for hypergraphs.

math.CO↗
arXiv2025

Rainbow Erd\H{o}s-S\'os Conjectures

An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a variant on the classical Tur\'an problem, asking for the maximum number of edges in a $n$-vertex properly edge-colored graph which does not contain a rainbow-$F$. In the following years many authors have studied the asymptotic behavior of $\mathrm{ex}^*(n,F)$ when $F$ is bipartite. In the particular case that $F$ is a tree $T$, the infamous Erd\"os-S\'os conjecture says that the extremal number of $T$ depends only on the size of $T$ and not its structure. After observing that such a pattern cannot hold for $\mathrm{ex}^*$ in the usual setting, we propose that the relative rainbow extremal number $\mathrm{ex}^*(Q_n,T)$ in the $n$-dimensional hypercube $Q_n$ will satisfy an Erd\"os-S\'os type Conjecture and verify it for some infinite families of trees $T$.

math.CO↗
arXiv2024

$k$-Hyperopic Cops and Robber

A generalization of hyperopic cops and robber, analogous to the $k$-visibility cops and robber, is introduced in this paper. For a positive integer $k$ the $k$-hyperopic game of cops and robber is defined similarly as the usual cops and robber game, but with the robber being omniscient and invisible to the cops that are at distance at most $k$ away from the robber. The cops win the game if, after a finite number of rounds, a cop occupies the same vertex as robber. Otherwise, robber wins. The minimum number of cops needed to win the game on a graph $G$ is the $k$-hyperopic cop number $c_{H,k}(G)$ of $G$. In addition to basic properties of the new invariant, cop-win graphs are characterized and a general upper bound in terms of the matching number of the graph is given. The invariant is also studied on trees where the upper bounds mostly depend on the relation between $k$ and the diameter of the tree. It is also proven that the 2-hyperopic cop number of outerplanar graphs is at most 2 and an upper bound in terms of the number of vertices of the graph is presented for $k \geq 3$.

math.CO↗
arXiv2024

Evacuation Planning on Time-Expanded Networks with Integrated Wildfire Information

We study the problem of evacuation planning for natural disasters, focusing on wildfire evacuations. By creating pre-planned evacuation routes that can be updated based on real-time data, we provide an easily adjustable approach to evacuation planning and implementation. Our method uses publicly available data and can be tailored for a particular region or circumstance. We formulate large-scale evacuations as maximum flow problems on time-expanded networks, in which we integrate hazard information given in the form of a shapefile. An initial flow and evacuation plan is found based on a predicted fire, and is then updated based on revised fire information received during the evacuation. We provide a proof of concept on three locations with historic deadly fires using data available through OpenStreetMaps, a basemap for a geographic information system (GIS), on a NetworkX Python script. The results validate viable running times and quality of information for application in practice. Particular strengths are the scalability and modularity of our approach and accompanying software package.

math.OC↗
arXiv2024

The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs

We introduce a new tool useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted $\chi_{\mathsf{o}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted $\chi_{\mathsf{PCF}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing uniquely in its neighborhood. Our new tool works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove $\chi_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $\chi_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.

math.CO↗
arXiv2023

On the interval coloring impropriety of graphs

An improper interval (edge) coloring of a graph $G$ is an assignment of colors to the edges of $G$ satisfying the condition that, for every vertex $v \in V(G)$, the set of colors assigned to the edges incident with $v$ forms an integral interval. An interval coloring is $k$-improper if at most $k$ edges with the same color all share a common endpoint. The minimum integer $k$ such that there exists a $k$-improper interval coloring of the graph $G$ is the interval coloring impropriety of $G$, denoted by $\mu_{int}(G)$. In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. This provides additional evidence to the conjecture by Casselgren and Petrosyan that $\mu_{int}(G)\leq 2$ for all complete multipartite graphs $G$. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, $G\odot H$.

math.CO↗
arXiv2021

Percolation transition for random forests in $d\geq 3$

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta>0$ per edge. It arises as the $q\to 0$ limit of the $q$-state random cluster model with $p=\beta q$. We prove that in dimensions $d\geq 3$ the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ where no percolation transition occurs. The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\mathbb{H}^{0|2}$. This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the $\mathbb{H}^{0|2}$ model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

math.PR↗
arXiv2021

3-colorability of graphs with minimum degree at least 6

Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where $n_i$ is the number of vertices in $G$ with list of size $i$ for $i\in \{2,3\}$. As a corollary, we conclude that the 3-colorability of any graph $G$ with minimum degree at least 6 can be determined in $O(1.3196^{n-.5Δ(G)})$ time.

math.CO↗
arXiv2020

Random spanning forests and hyperbolic symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $β>0$ per edge. This is called the arboreal gas model, and the special case when $β=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=β/(1+β)$ conditioned to be acyclic, or as the limit $q\to 0$ with $p=βq$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $β=α/N$ there is a phase transition similar to that of the Erdős--Rényi random graph: a giant tree percolates for $α> 1$ and all trees have bounded size for $α<1$. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $\mathbb{Z}^2$ for any finite $β>0$. This result is a consequence of a Mermin--Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

math.PR↗
arXiv2020

Macroscopic loops in the loop O(n) model via the XOR trick

The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model: - In the regime $(n,x) \in [1,1+\delta) \times (1- \delta, 1]$ with $\delta >0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+\delta)$ and $x \in (1-\delta,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. - Existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - Existence of non-contractible loops on a torus when $n \in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $\omega$ is a collection of short loops and $\Gamma$ is a long loop, then the symmetric difference of $\omega$ and $\Gamma$ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs.

math.PR↗
arXiv2018

Eigenvector correlations in the complex Ginibre ensemble

The complex Ginibre ensemble is an $N\times N$ non-Hermitian random matrix over $\mathbb{C}$ with i.i.d. complex Gaussian entries normalized to have mean zero and variance $1/N$. Unlike the Gaussian unitary ensemble, for which the eigenvectors are distributed according to Haar measure on the compact group $U(N)$, independently of the eigenvalues, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. In this paper we systematically study properties of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes their asymptotic behavior (as $N$ goes to infinity). Our work extends previous results of Chalker and Mehlig [CM98], in which the correlation for pairs of eigenvectors was computed.

math.PR↗
arXiv2016

Emptiness Formation Probability

We present rigorous upper and lower bounds on the emptiness formation probability for the ground state of a spin-$1/2$ Heisenberg XXZ quantum spin system. For a $d$-dimensional system we find a rate of decay of the order $\exp(-c L^{d+1})$ where $L$ is the sidelength of the box in which we ask for the emptiness formation event to occur. In the $d=1$ case this confirms previous predictions made in the integrable systems community, though our bounds do not achieve the precision predicted by Bethe ansatz calculations. On the other hand, our bounds in the case $d \geq 2$ are new. The main tools we use are reflection positivity and a rigorous path integral expansion which is a variation on those previously introduced by Toth, Aizenman-Nachtergaele and Ueltschi.

math-ph↗
arXiv2015

Uniqueness regime for Markov dynamics on quantum lattice spin systems

We consider a lattice of weakly interacting quantum Markov processes. Without interaction, the dynamics at each site is relaxing exponentially to a unique stationary state. With interaction, we show that there remains a unique stationary state in the thermodynamic limit, i.e. absence of phase coexistence, and the relaxation towards it is exponentially fast for local observables. We do not assume that the quantum Markov process is reversible (detailed balance) w.r.t. a local Hamiltonian.

math-ph↗
arXiv2012

Random Field Induced Order in Low Dimension

Consider the behavior of a classical O(n) model in a weak random external field acting along some $k$-dimensional subspace in $\R^n$ with $k<n$. We show rigorously that if $k=n-1$, for the model defined on $\Z^d$, $d ={2, 3}$ there is residual magnetic order perpendicular to the subspace which supports the distribution of the random field only. Furthermore, when $k\leq n-1$ and $d\geq 2$ we show in general that the magnitude of spin projections onto this subspace are small.

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