arXiv subjects

John Sylvester

Publications and source records attributed to John Sylvester.

At least 19 recordsLinked to original sources

arXiv2026

Set-defined graph classes: $χ$-boundedness meets tropical algebra

We study set-defined graph classes: hereditary classes whose vertices are assigned fixed-length numerical tuples, with adjacency determined solely by equality patterns among coordinates. These classes arise in structural graph theory, communication complexity, logic, and adjacency labeling schemes. We ask when they are $χ$-bounded, that is, when chromatic number is bounded in terms of clique number throughout the class. First, we prove a decomposition theorem: every graph in a set-defined class can be partitioned into a number of parts polynomially bounded in its clique number, each inducing a union of a bounded number of shift-colorable graphs, that is, graphs admitting a homomorphism to a shift graph. Thus bounded unions of shift-colorable graphs form the fundamental obstruction to $χ$-boundedness in set-defined classes. For full set-defined classes, consisting of all graphs realizable by a fixed Boolean rule on equality patterns, we prove a stronger dichotomy: every such class is either polynomially $χ$-bounded or contains shift graphs of arbitrarily large chromatic number. Moreover, we provide an algorithm that, given a Boolean-function description of a full set-defined class, decides $χ$-boundedness of the class. It reduces the problem to feasibility of tropical linear programs, and its correctness follows from a duality with winning strategies in mean-payoff games. Conversely, every integer system of tropical inequalities, and hence every mean-payoff game, can be encoded in strongly polynomial time as a set-defined class whose non-$χ$-boundedness is equivalent to feasibility. This provides a graph-theoretic counterpart of tropical feasibility and mean-payoff-game solvability, linking structural graph theory, tropical algebra, and game-theoretic algorithms.

cs.DM↗

arXiv2025

Capturing an Invisible Robber using Separators

We study the zero-visibility cops and robbers game, where the robber is invisible to the cops until they are caught. This differs from the classic game where full information about the robber's location is known at any time. A previously known solution for capturing a robber in the zero-visibility case is based on the pathwidth decomposition. We provide an alternative solution based on a separation hierarchy, improving capture time and space complexity without asymptotically increasing the zero-visibility cop number in most cases. In addition, we provide a better bound on the approximate zero-visibility cop number for various classes of graphs, where approximate refers to the restriction to polynomial time computable strategies.

cs.DM↗

arXiv2025

Temporal Exploration of Random Spanning Tree Models

The Temporal Graph Exploration problem (TEXP) takes as input a temporal graph, i.e., a sequence of graphs $(G_i)_{i\in \mathbb{N}}$ on the same vertex set, and asks for a walk of shortest length visiting all vertices, where the $i$-th step uses an edge from $G_i$. If each such $G_i$ is connected, then an exploration of length $n^2$ exists, and this is known to be the best possible up to a constant. More fine-grained lower and upper bounds have been obtained for restricted temporal graph classes, however, for several fundamental classes, a large gap persists between known bounds, and it remains unclear which properties of a temporal graph make it inherently difficult to explore. Motivated by this limited understanding and the central role of the Temporal Graph Exploration problem in temporal graph theory, we study the problem in a randomised setting. We introduce the Random Spanning Tree (RST) model, which consists of a set of $n$-vertex trees together with an arbitrary probability distribution $μ$ over this set. A random temporal graph generated by the RST model is a sequence of independent samples drawn from $μ$. We initiate a systematic study of the Temporal Graph Exploration problem in such random temporal graphs and establish tight general bounds on exploration time. Our first main result proves that any RST model can, with high probability (w.h.p.), be explored in $O(n^{3/2})$ time, and we show that this bound is tight up to a constant factor. This demonstrates a fundamental difference between the adversarial and random settings. Our second main result shows that if all trees of an RST are subgraphs of a fixed graph with $m$ edges then, w.h.p.\ , it can be explored in $O(m)$ time.

cs.DM↗

arXiv2025

Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with parameter $0<q(n)\leq 1$. This random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path, and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.

math.CO↗

arXiv2024

Boolean combinations of graphs

Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and limitations of boolean combinations in this context, we investigate how they affect different combinatorial and structural properties of graphs, in particular $χ$-boundedness, as well as characterize the structure of boolean combinations of graphs from various classes.

math.CO↗

arXiv2024

Functionality of Random Graphs

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph $G(n,p)$ up to a constant factor for every value of $p$.

math.CO↗

arXiv2024

Time-Biased Random Walks and Robustness of Expanders

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral gap $γ$ (asymptotically) bounded away from $0$. Our work has two main strands. First, we establish a dichotomy for the robustness of mixing times on edge-weighted $d$-regular graphs (i.e., reversible Markov chains) subject to a Lipschitz condition, which bounds the ratio of adjacent weights by $β\geq 1$. If $β\ge 1$ is sufficiently small, then $γ\asymp 1$ and the mixing time is logarithmic in $n$. On the other hand, if $β\geq 2d$, there is an edge-weighting such that $γ$ is polynomially small in $1/n$. Second, we apply our robustness result to a time-dependent version of the so-called $\varepsilon$-biased random walk, as introduced in Azar et al. [Combinatorica 1996]. We show that, for any constant $\varepsilon>0$, a bias strategy can be chosen adaptively so that the $\varepsilon$-biased random walk covers any bounded-degree regular expander in $Θ(n)$ expected time, improving the previous-best bound of $O(n \log \log n)$. We prove the first non-trivial lower bound on the cover time of the $\varepsilon$-biased random walk, showing that, on bounded-degree regular expanders, it is $ω(n)$ whenever $\varepsilon = o(1)$. We establish this by controlling how much the probability of arbitrary events can be ``boosted'' by using a time-dependent bias strategy.

math.PR↗

arXiv2024

Adjacency Labeling Schemes for Small Classes

A graph class admits an implicit representation if, for every positive integer $n$, its $n$-vertex graphs have a $O(\log n)$-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length $O(\log n)$ such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing $2^{O(n \log n)}$ $n$-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami and Hatami, FOCS '22], and does not even hold among monotone (i.e., closed under taking subgraphs) factorial classes [Bonnet et al., ICALP '24]. However, monotone small (i.e., containing at most $n! c^n$ many $n$-vertex graphs for some constant $c$) classes do admit implicit representations. This motivates the Small Implicit Graph Conjecture: Every hereditary small class admits an $O(\log n)$-bit labeling scheme. We provide evidence supporting the Small Implicit Graph Conjecture. First, we show that every small weakly sparse (i.e., excluding some fixed bipartite complete graph as a subgraph) class has an implicit representation. This is a consequence of the following fact of independent interest proved in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). Second, we show that every hereditary small class admits an $O(\log^3 n)$-bit labeling scheme, which provides a substantial improvement of the best-known polynomial upper bound of $n^{1-\varepsilon}$ on the size of adjacency labeling schemes for such classes. This is a consequence of another fact of independent interest proved in the paper: Every small class has neighborhood complexity $O(n \log n)$.

math.CO↗

arXiv2024

Coupled transport equations with freezing

We study a system of two coupled transport equations with freezing. The solutions freeze in time when they are equal. We prove existence and uniqueness of continuous solutions if the initial conditions are continuous. We discuss several qualitative and quantitative properties of the solutions. The equations arise in a model for collisions of a large number of tightly spaced balls.

math.AP↗

arXiv2024

Symmetric-Difference (Degeneracy) and Signed Tree Models

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most $d$ if it admits an elimination order of its vertices where a vertex $u$ can be removed whenever it has a $d$-twin, i.e., another vertex $v$ such that at most $d$ vertices outside $\{u,v\}$ are neighbors of exactly one of $u, v$. The family of graph classes of bounded sd-degeneracy is a superset of that of graph classes of bounded degeneracy or of bounded flip-width, and more generally, of bounded symmetric difference. Unlike most graph parameters, sd-degeneracy is not hereditary: it may be strictly smaller on a graph than on some of its induced subgraphs. In particular, every $n$-vertex graph is an induced subgraph of some $O(n^2)$-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise $\tilde{O}(\sqrt n)$-bit adjacency labeling schemes for them, which are optimal up to the hidden polylogarithmic factor. This is attained on some even more general classes, consisting of graphs $G$ whose vertices bijectively map to the leaves of a tree $T$, where transversal edges and anti-edges added to $T$ define the edge set of $G$. We call such graph representations signed tree models as they extend the so-called tree models (or twin-decompositions) developed in the context of twin-width, by adding transversal anti-edges. While computing the degeneracy of an input graph can be done in linear time, we show that deciding whether its symmetric difference is at most 8 is co-NP-complete, and whether its sd-degeneracy is at most 1 is NP-complete.

cs.DS↗

arXiv2024

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.

cs.DM↗

arXiv2024

Cover and Hitting Times of Hyperbolic Random Graphs

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.

math.PR↗

arXiv2024

Tight bounds on adjacency labels for monotone graph classes

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined solely from their labels. We give tight bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a well-behaved growth function between $2^{O(n \log n)}$ and $2^{O(n^{2-δ})}$ for any $δ> 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-δ}$ for any fixed $δ> 0$, and some~sub-multiplicativity condition, there are monotone graph classes with growth $2^{O(nf(n))}$ that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand, any such class does admit adjacency labels of size $O(f(n)\log n)$. Surprisingly this tight bound is a $Θ(\log n)$ factor away from the information-theoretic bound of $Ω(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone \emph{small} classes. In other words, any monotone class with growth rate at most $n!\,c^n$ for some constant $c>0$, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy.We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.

math.CO↗

arXiv2024

Small But Unwieldy: A Lower Bound on Adjacency Labels for Small Classes

We show that for any natural number $s$, there is a constant $γ$ and a subgraph-closed class having, for any natural $n$, at most $γ^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most $s \log n$. In other words, for every $s$, there is a small (even tiny) monotone class without universal graphs of size $n^s$. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size $(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmović et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.

math.CO↗

arXiv2024

Mean-Biased Processes for Balanced Allocations

We introduce a new class of balanced allocation processes which bias towards underloaded bins (those with load below the mean load) either by skewing the probability by which a bin is chosen for an allocation (probability bias), or alternatively, by adding more balls to an underloaded bin (weight bias). A prototypical process satisfying the probability bias condition is Mean-Thinning: At each round, we sample one bin and if it is underloaded, we allocate one ball; otherwise, we allocate one ball to a second bin sample. Versions of this process have been in use since at least 1986. An example of a process, introduced by us, which satisfies the weight bias condition is Twinning: At each round, we only sample one bin. If the bin is underloaded, then we allocate two balls; otherwise, we allocate only one ball. Our main result is that for any process with a probability or weight bias, with high probability the gap between maximum and minimum load is logarithmic in the number of bins. This result holds for any number of allocated balls (heavily loaded case), covers many natural processes that relax the Two-Choice process, and we also prove it is tight for many such processes, including Mean-Thinning and Twinning. Our analysis employs a delicate interplay between linear, quadratic and exponential potential functions. It also hinges on a phenomenon we call "mean quantile stabilization", which holds in greater generality than our framework and may be of independent interest.

math.PR↗

arXiv2023

Rumors with Changing Credibility

Randomized rumor spreading processes diffuse information on an undirected graph and have been widely studied. In this work, we present a generic framework for analyzing a broad class of such processes on regular graphs. Our analysis is protocol-agnostic, as it only requires the expected proportion of newly informed vertices in each round to be bounded, and a natural negative correlation property. This framework allows us to analyze various protocols, including PUSH, PULL, and PUSH-PULL, thereby extending prior research. Unlike previous work, our framework accommodates message failures at any time $t\geq 0$ with a probability of $1-q(t)$, where the credibility $q(t)$ is any function of time. This enables us to model real-world scenarios in which the transmissibility of rumors may fluctuate, as seen in the spread of ``fake news'' and viruses. Additionally, our framework is sufficiently broad to cover dynamic graphs.

cs.DM↗

arXiv2023

Cops and Robbers on Multi-Layer Graphs

We generalise the popular cops and robbers game to multi-layer graphs, where each cop and the robber are restricted to a single layer (or set of edges). We show that initial intuition about the best way to allocate cops to layers is not always correct, and prove that the multi-layer cop number is neither bounded from above nor below by any increasing function of the cop numbers of the individual layers. We determine that it is NP-hard to decide if $k$ cops are sufficient to catch the robber, even if every cop layer is a tree and a set of isolated vertices. However, we give a polynomial time algorithm to determine if $k$ cops can win when the robber layer is a tree. Additionally, we investigate a question of worst-case divisions of a simple graph into layers: given a simple graph $G$, what is the maximum number of cops required to catch a robber over all multi-layer graphs where each edge of $G$ is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we determine this parameter up to multiplicative constants. Lastly we consider a multi-layer variant of Meyniel's conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times $n / \log n$, where $n$ is the number of vertices in the graph.

math.CO↗

arXiv2023

The Power of Filling in Balanced Allocations

We introduce a new class of balanced allocation processes which are primarily characterized by ``filling'' underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is $\mathcal{O}(\log n)$ w.h.p. for any number of balls $m\geq 1$. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample-efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of $\mathcal{O}(\log n)$ on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar and Shah (2002).

cs.DM↗