Graphs Of Degree Research Articles

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.

Finding dense minors using average degree

AbstractMotivated by Hadwiger's conjecture, we study the problem of finding the densest possible ‐vertex minor in graphs of average degree at least . We show that if has average degree at least , it contains a minor on vertices with at least edges. We show that this cannot be improved beyond . Finally, for we exactly determine the number of edges we are guaranteed to find in the densest ‐vertex minor in graphs of average degree at least .

Graphs of Degree at Least \({3}\) with Minimum Algebraic Connectivity

Graphs of Degree at Least \({3}\) with Minimum Algebraic Connectivity

A family of graphs that cannot occur as character degree graphs of solvable groups

A family of graphs that cannot occur as character degree graphs of solvable groups

Level-set percolation of Gaussian random fields on complex networks.

We provide an explicit solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. The solution requires an analysis of the heterogeneous microstructure of the percolation problem, i.e., a self-consistent determination of locally varying percolation probabilities. This is achieved using a cavity or message passing approach. It can be evaluated, both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class.

DDGPnP: Differential degree graph based PnP solution to handle outliers

Existing external relationships for outlier removal in the perspective-n-point problem are generally spatial coherence among the neighbor correspondences. In the situation of high noise or spatially incoherent distributions, pose estimation is relatively inaccurate due to a small number of detected inliers. To address these problems, this paper explores the globally coherent external relationships for outlier removal and pose estimation. To this end, the differential degree graph (DDG) is proposed to employ the intersection angles between rays of correspondences to handle outliers. Firstly, a pair of two degree graphs are constructed to establish the external relationships between 3D-2D correspondences in the world and camera coordinates. Secondly, the DDG is estimated through subtracting the two degree graphs and operating binary operation with a degree threshold. Besides, this paper mathematically proves that the maximum clique of the DDG represents the inliers. Thirdly, a novel vertice degree based method is put forward to extract the maximum clique from DDG for outlier removal. Besides, this paper proposes a novel pipeline of DDG based PnP solution, i.e. DDGPnP, to achieve accurate pose estimation. Experiments demonstrate the superiority and effectiveness of the proposed method in the aspects of outlier removal and pose estimation by comparison with the state of the arts. Especially for the high noise situation, the DDGPnP method can achieve not only accurate pose but also a large number of correct correspondences.

On vertex‐transitive graphs with a unique hamiltonian cycle

AbstractA graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex‐transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.

Reducing linear Hadwiger’s conjecture to coloring small graphs

In 1943, Hadwiger conjectured that every graph with no K t K_t minor is ( t − 1 ) (t-1) -colorable for every t ≥ 1 t\ge 1 . In the 1980s, Kostochka and Thomason independently proved that every graph with no K t K_t minor has average degree O ( t log ⁡ t ) O(t\sqrt {\log t}) and hence is O ( t log ⁡ t ) O(t\sqrt {\log t}) -colorable. Recently, Norin, Song and the second author showed that every graph with no K t K_t minor is O ( t ( log ⁡ t ) β ) O(t(\log t)^{\beta }) -colorable for every β > 1 / 4 \beta > 1/4 , making the first improvement on the order of magnitude of the O ( t log ⁡ t ) O(t\sqrt {\log t}) bound. The first main result of this paper is that every graph with no K t K_t minor is O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. This is actually a corollary of our main technical result that the chromatic number of a K t K_t -minor-free graph is bounded by O ( t ( 1 + f ( G , t ) ) ) O(t(1+f(G,t))) where f ( G , t ) f(G,t) is the maximum of χ ( H ) a \frac {\chi (H)}{a} over all a ≥ t log ⁡ t a\ge \frac {t}{\sqrt {\log t}} and K a K_a -minor-free subgraphs H H of G G that are small (i.e. O ( a log 4 ⁡ a ) O(a\log ^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small K t K_t -minor-free graphs, we show that K t K_t -minor-free graphs are O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. Second, it shows that proving Linear Hadwiger’s Conjecture (that K t K_t -minor-free graphs are O ( t ) O(t) -colorable) reduces to proving it for small graphs. Third, we prove that K t K_t -minor-free graphs with clique number at most log ⁡ t / ( log ⁡ log ⁡ t ) 2 \sqrt {\log t}/ (\log \log t)^2 are O ( t ) O(t) -colorable. This implies our final corollary that Linear Hadwiger’s Conjecture holds for K r K_r -free graphs for every fixed r r ; more generally, we show there exists C ≥ 1 C\ge 1 such that for every r ≥ 1 r\ge 1 , there exists t r t_r such that for all t ≥ t r t\ge t_r , every K r K_r -free K t K_t -minor-free graph is C t Ct -colorable. One key to proving the main theorem is building the minor in two new ways according to whether the chromatic number ‘separates’ in the graph: sequentially if the graph is the chromatic-inseparable and recursively if the graph is chromatic-separable. The other key is a new standalone result that every K t K_t -minor-free graph of average degree d = Ω ( t ) d=\Omega (t) has a subgraph on O ( t log 3 ⁡ t ) O(t \log ^3 t) vertices with average degree Ω ( d ) \Omega (d) .

Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.

Statistical physics of principal minors: Cavity approach.

Determinants are useful to represent the state of an interacting system of (effectively) repulsive and independent elements, like fermions in a quantum system and training samples in a learning problem. A computationally challenging problem is to compute the sum of powers of principal minors of a matrix which is relevant to the study of critical behaviors in quantum fermionic systems and finding a subset of maximally informative training data for a learning algorithm. Specifically, principal minors of positive square matrices can be considered as statistical weights of a random point process on the set of the matrix indices. The probability of each subset of the indices is in general proportional to a positive power of the determinant of the associated submatrix. We use Gaussian representation of the determinants for symmetric and positive matrices to estimate the partition function (or free energy) and the entropy of principal minors within the Bethe approximation. The results are expected to be asymptotically exact for diagonally dominant matrices with locally treelike structures. We consider the Laplacian matrix of random regular graphs of degree K=2,3,4 and exactly characterize the structure of the relevant minors in a mean-field model of such matrices. No (finite-temperature) phase transition is observed in this class of diagonally dominant matrices by increasing the positive power of the principal minors, which here plays the role of an inverse temperature.

On the complexity of distance-d independent set reconfiguration

For a fixed positive integer d≥2, a distance-d independent set (DdIS) of a graph is a vertex-subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (TS) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (TJ), the target vertex needs not to be adjacent to the original one. The Distance-dIndependent Set Reconfiguration (DdISR) problem under TS/TJ asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d=2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under TS and TJ for any fixed d≥3. On chordal graphs, we show that DdISR under TJ is in P when d is even and PSPACE-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is PSPACE-complete for d=2 but in P for d=3 under TS, while under TJ it is in P for d=2 but PSPACE-complete for d=3. Additionally, certain well-known hardness results for d=2 on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for d≥3.

Extensions of results on phylogeny graphs of degree bounded digraphs

An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i,j) digraph for some positive integers i and j. The phylogeny graph of a digraph D has V(D) as the vertex set and an edge uv if and only if one of the following is true: (u,v)∈A(D); (v,u)∈A(D); (u,w)∈A(D) and (v,w)∈A(D) for some w∈V(D). A graph G is a phylogeny graph (resp. an (i,j) phylogeny graph) if there is an acyclic digraph D (resp. an (i,j) digraph D) such that the phylogeny graph of D is isomorphic to G. Lee et al. (2017) and Eoh and Kim (2021) studied the (2,2) phylogeny graphs, (1,j) phylogeny graphs, (i,1) phylogeny graphs, and (2,j) phylogeny graphs. Their work was motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have chordal moral graphs (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work by giving necessary conditions of chordal (i,2) phylogeny graphs. We go further to give necessary conditions of (i,j) phylogeny graphs by listing forbidden induced subgraphs.

Robust extended states in Anderson model on partially disordered random regular graphs

In this work we analytically explain the origin of the mobility edge in the partially disordered random regular graphs of degree d, i.e., with a fraction \betaβ of the sites being disordered, while the rest remain clean. It is shown that the mobility edge in the spectrum survives in a certain range of parameters (d,\betaβ) at infinitely large uniformly distributed disorder. The critical curve separating extended and localized states is derived analytically and confirmed numerically. The duality in the localization properties between the sparse and extremely dense RRG has been found and understood.

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 $q$ .

On the complexity of nucleolus computation for bipartite b-matching games

We explore the complexity of nucleolus computation in b-matching games on bipartite graphs. We show that computing the nucleolus of a simple b-matching game is NP-hard when b≡3 even on bipartite graphs of maximum degree 7. We complement this with partial positive results in the special case where b values are bounded by 2. In particular, we describe an efficient algorithm when a constant number of vertices satisfy bv=2 as well as an efficient algorithm for computing the non-simple b-matching nucleolus when b≡2.

Polyhedra without cubic vertices are prism‐hamiltonian

AbstractThe prism over a graph is the Cartesian product of with the complete graph on two vertices. A graph is prism‐hamiltonian if the prism over is hamiltonian. We prove that every polyhedral graph (i.e., 3‐connected planar graph) of minimum degree at least four is prism‐hamiltonian.

The overfull conjecture on graphs of odd order and large minimum degree

AbstractLet be a simple graph with maximum degree . A subgraph of is overfull if . Chetwynd and Hilton in 1986 conjectured that a graph with has chromatic index if and only if contains no overfull subgraph. Let , be sufficiently large, and be graph on vertices with minimum degree at least . It was shown that the conjecture holds for if is even. In this paper, the same result is proved if is odd. As far as we know, this is the first result on the Overfull Conjecture for graphs of odd order and with a minimum degree constraint.

Sum labelling graphs of maximum degree two

The concept of sum labelling was introduced in 1990 by Harary. A graph is a sum graph if its vertices can be labelled by distinct positive integers in such a way that two vertices are connected by an edge if and only if the sum of their labels is the label of another vertex in the graph. It is easy to see that every sum graph has at least one isolated vertex, and every graph can be made a sum graph by adding at most n2 isolated vertices to it. The minimum number of isolated vertices that need to be added to a graph to make it a sum graph is called the sum number of the graph.The sum number of several prominent graph classes (e.g., cycles, trees, complete graphs) is already well known. We examine the effect of taking the disjoint union of graphs on the sum number. In particular, we provide a complete characterisation of the sum number of graphs of maximum degree two, since every such graph is the disjoint union of paths and cycles.

Eulerian character degree graphs of solvable groups

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in Irr ( G ) . Let ρ ( G ) be the set of all primes that divide some degrees in cd ( G ) . The character degree graph Δ ( G ) of G is the simple undirected graph with vertex set ρ ( G ) and in which two distinct vertices p and q are adjacent if there exists a character degree r ∈ cd ( G ) such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph Δ ( G ) of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and n ≥ 4 is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.