Bounded Degree Graphs Research Articles

First passage percolation with recovery

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph G place a red particle at a reference vertex o and colorless particles (seeds) at all other vertices. The red particle starts spreading a red first passage percolation of rate 1, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading red first passage percolation. All vertices are equipped with independent exponential clocks ringing at rate γ>0, when a clock rings the corresponding red vertex turns black. For t≥0, let Ht and Mt denote the size of the longest red path and of the largest red cluster present at time t. If G is the semi-line, then for all γ>0 almost surely lim suptHtloglogtlogt=1 and lim inftHt=0. In contrast, if G is an infinite Galton–Watson tree with offspring mean m>1 then, for all γ>0, almost surely lim inftHtlogtt≥m−1 and lim inftMtloglogtt≥m−1, while lim suptMtect≤1, for all c>m−1. Also, almost surely as t→∞, for all γ>0Ht is of order at most t. Furthermore, if we restrict our attention to bounded-degree graphs, then for any ɛ>0 there is a critical value γc>0 so that for all γ>γc, almost surely lim suptMtt≤ɛ.

List packing number of bounded degree graphs

Abstract We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $\Delta$ has list packing number at most $\Delta +1$ . Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.

The matroid of a graphing

Graphings serve as limit objects for bounded-degree graphs. We define the “cycle matroid” of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We prove that for a Benjamini–Schramm convergent sequence of graphs, the total rank, normalized by the number of nodes, converges to the total rank of the limit graphing.

On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing

On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing

Ramsey numbers for multiple copies of sparse graphs

AbstractFor a graph and an integer , we let denote the disjoint union of copies of . In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for , one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant such that , provided is sufficiently large. Subsequently, Burr gave an implicit way of computing and noted that this long‐term behaviour occurs when is triply exponential in . Very recently, Bucić and Sudakov revived the problem and established an essentially tight bound on by showing follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on in case is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state‐of‐the‐art bounds on and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rödl and Ruciński, with an emphasis on developing an efficient absorbing method for bounded degree graphs.

Algorithms for the ferromagnetic Potts model on expanders

AbstractWe give algorithms for approximating the partition function of the ferromagnetic $q$ -color Potts model on graphs of maximum degree $d$ . Our primary contribution is a fully polynomial-time approximation scheme for $d$ -regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger’s algorithm to count cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high-temperature proof follows more standard polymer model analysis, our result holds in the largest-known range of parameters $d$ and $q$ .

Fast Sampling via Spectral Independence Beyond Bounded-degree Graphs

Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.

Fast and Perfect Sampling of Subgraphs and Polymer Systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets ) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

The Limit of the Zero Locus of the Independence Polynomial for Bounded Degree Graphs

The Limit of the Zero Locus of the Independence Polynomial for Bounded Degree Graphs

Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

Distributed half-integral matching and beyond

By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log⁎⁡n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from {0,12,1}. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree Δ=2d, and any distributed graph algorithm with round complexity T(Δ) that only depends on Δ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2d. We give a new algorithm that shows that this is also sufficient.

Algorithmic Aspects of Total Vertex-Edge Domination in Graphs

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

On the zeroes of hypergraph independence polynomials

AbstractWe study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta )$ ). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$ . We conjecture that for $k\ge 3$ , $k$ -uniform linear hypergraphs have a much larger zero-free disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.

Hardness results of global roman domination in graphs

A Roman dominating function (RDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. A global Roman dominating function (GRDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that f is a Roman dominating function of both G and its complement G¯. The weight of f is f(V)=Σu∈Vf(u). The minimum weight of a GRDF in a graph G is known as global Roman domination number of G and is denoted by γgR(G). Minimum Global Roman Domination is to find a global Roman dominating function of minimum weight and Decide Global Roman Domination is the decision version of Minimum Global Roman Domination. In this paper, we show that Decide Global Roman Domination is NP-complete for bipartite graphs and chordal graphs. On the positive side, we give a polynomial-time algorithm for Minimum Global Roman Domination for threshold graphs. Further, we propose an O(ln|V|)-approximation algorithm for Minimum Global Roman Domination for a graph G=(V,E). We also show that Minimum Global Roman Domination cannot be approximated within a factor of (1−ϵ)ln|V| for any ϵ>0 unless P = NP. Next, we show that Minimum Global Roman Domination admits a constant approximation algorithm for bounded degree graphs. Finally, we show that Minimum Global Roman Domination is APX-complete for bounded degree graphs.

On Bounded Degree Graphs with Large Size-Ramsey Numbers

On Bounded Degree Graphs with Large Size-Ramsey Numbers

Node and edge averaged complexities of local graph problems

We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph G=(V,E) is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree Delta is at least Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big ). This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big )—this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an O(sqrt{log log n}) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big ), the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matching is O(log ^2Delta + log ^* n) and the deterministic node-averaged complexity of maximal matching is O(log ^3Delta + log ^* n). Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be Theta (log n), even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity O(log ^* n), while keeping the worst-case complexity in O(log n).

Upgrading MLSI to LSI for reversible Markov chains

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by log⁡(1/p), where p is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobolev estimate for Zero-Range processes on arbitrary graphs. As another application, we determine the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs, and use it to provide negative answers to two open questions by Montenegro and Tetali (2006) [27] and Hermon and Peres (2018) [17]. Our proof builds upon the ‘regularization trick’ recently introduced by the last two authors.

Hardness Results of Connected Power Domination for Bipartite Graphs and Chordal Graphs

A set [Formula: see text] of a graph [Formula: see text] is called a connected power dominating set of [Formula: see text] if [Formula: see text], the subgraph induced by [Formula: see text], is connected and every vertex in the graph can be observed from [Formula: see text], following the two observation rules for power system monitoring: Rule [Formula: see text]: if [Formula: see text], then [Formula: see text] can observe itself and all its neighbors, and Rule [Formula: see text]: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. Given a graph [Formula: see text], Minimum Connected Power Domination is to find a connected power dominating set of minimum cardinality of [Formula: see text] and Decide Connected Power Domination is the decision version of Minimum Connected Power Domination. Decide Connected Power Domination is known to be NP -complete for general graphs. In this paper, we prove that Decide Connected Power Domination remains NP -complete for star-convex bipartite graphs, perfect elimination bipartite graphs and split graphs. This answers some open problems posed in [B. Brimkov, D. Mikesell and L. Smith, Connected power domination in graphs, J. Comb. Optim. 38(1) (2019) 292–315]. On the positive side, we show that Minimum Connected Power Domination is polynomial-time solvable for chain graphs, a proper subclass of perfect elimination bipartite graph, and for threshold graphs, a proper subclass of split graphs. Further, we show that Minimum Connected Power Domination cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text], for bipartite graphs as well as for chordal graphs. Finally, we show that Minimum Connected Power Domination is APX -hard for bounded degree graphs.

Edge and Pair Queries-Random Graphs and Complexity

We investigate two types of query games played on a graph, pair queries and edge queries. We concentrate on investigating the two associated graph parameters for binomial random graphs, and showing that determining any of the two parameters is NP-hard for bounded degree graphs.

Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

Let $D = d_1, d_2, \ldots, d_n$ and $F = f_1, f_2,\ldots, f_n$ be two sequences of positive integers. We consider the following decision problems: is there a $i)$ multigraph, $ii)$ loopless multigraph, $iii)$ simple graph, $iv)$ connected simple graph, $v)$ tree, $vi)$ caterpillar $G = (V,E)$ such that for all $k$, $d(v_k) = d_k$ and $\sum_{w\in \mathcal{N}(v_k)} d(w) = f_k$ ($d(v)$ is the degree of $v$ and $\mathcal{N}(v)$ is the set of neighbors of $v$). Here we show that all these decision problems can be solved in polynomial time if $\max_{k} d_k$ is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.