pinecone/core-2020-05-10-deduplication huggingface dataset:
A Pattern of Asymptotic Vertex Valency Distributions in Planar Maps

AbstractLet a vertex be selected at random in a set ofn-edged rooted planar maps andpkdenote the limit probability (asn→∞) of this vertex to be of valencyk. For diverse classes of maps including Eulerian, arbitrary, polyhedral, and loopless maps as well as 2- and 3-connected triangulations, it is shown that non-zeropkbehave asymptotically in auniformmanner:pk∼c(πk)−1/2rkask→∞ with some constantsrandcdepending on the class. This distribution pattern can be reformulated in terms of the root vertex valency. By contrast,pk=2−kfor the class of arbitrary plane trees andpk=(k−1)2−kfor triangular dissections of convex polygons
A Description of Claw-Free Perfect Graphs

AbstractIt is known that all claw-free perfect graphs can be decomposed via clique-cutsets into two types of indecomposable graphs respectively called elementary and peculiar (1988, V. Chvátal and N. Sbihi,J. Combin. Theory Ser. B44, 154–176). We show here that every elementary graph is made up in a well-defined way of a line-graph of bipartite graph and some local augments consisting of complements of bipartite graphs. This yields a complete description of the structure of claw-free Berge graphs and a new proof of their perfectness
Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

AbstractIn this paper we study finite, connected, 4-valent graphsXwhich admit an action of a groupGwhich is transitive on vertices and edges, but not transitive on the arcs ofX. Such a graphXis said to be (G,1/2)-transitive. The groupGinduces an orientation of the edges ofX, and a certain class of cycles ofX(called alternating cycles) determined by the groupGis identified as having an important influence on the structure ofX. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown thatXis a cover of a finite, connected, 4-valent, (G,1/2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they aretightly attached, loosely attached, orantipodally attached.We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples
Some Formulas for Spin Models on Distance-Regular Graphs

AbstractA spin model is a square matrixWsatisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin modelWof an association scheme which containsWin its Bose–Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose–Mesner algebraN(W). In this paper we study the caseW∈A⊆N(W), where A is the Bose–Mesner algebra of a distance-regular graph. We show the following results. LetΓ=(X,R) be a distance-regular graph of diameterd>1 such that the Bose–Mesner algebra A ofΓsatisfiesW∈A⊆N(W) for some spin modelWonX. WriteW=∑di=0tiAi, whereAidenotes theith adjacency matrix. Setxi=t−1i−1tiandp=x−11x2. Thenxi=pi−1x1holds for alli. Moreover, the eigenvalues and the intersection numbers ofΓare rational functions ofx1andp
Distance Graphs andT-Coloring

AbstractWe discuss relationships amongT-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral setTthat contains 0, the asymptoticT-coloring ratioR(T) is equal to the fractional chromatic number of the distance graphG(Z,D), whereD=T−{0}. This fact is then used to study the distance graphs with distance sets of the formDm,k={1,2,…,m}−{k}. The chromatic numbers and the fractional chromatic numbers ofG(Z,Dm,k) are determined for all values ofmandk. Furthermore, circular chromatic numbers ofG(Z,Dm,k) for some special values ofmandkare obtained
Decompositions to Degree-Constrainded Subgraphs Are Simply Reducible to Edge-Colorings

AbstractThe degree-constrained subgraphs decomposition problem, such as anf-coloring, anf-factorization, and a [g,f]-factorization, is to decompose a given graphG=(V,E) to edge-disjoint subgraphs degree-constrained by integer-valued functionsfandgonV. In this paper we show that the problem can be simply reduced to the edge-coloring problem in polynomial-time. That is, for any positive integerk, we give a polynomial-time transformation ofGto a new graph such thatGcan be decomposed to at mostkdegree-constrained subgraphs if and only if the new graph can be edge-colored withkcolors
Locally Pancyclic Graphs

AbstractWe prove the following theorem. LetGbe a graph of ordernand letW⊆V(G). If |W|⩾3 anddG(x)+dG(y)⩾nfor every pair of non-adjacent verticesx, y∈W, then eitherGcontains cyclesC3, C4, …, C|W|such thatCicontains exactlyivertices fromW(i=3, 4, …, |W|), or |W|=nandG=Kn/2, n/2, or else |W|=4,G[W]=K2, 2. This generalizes a result of J. A. Bondy (1971,J. Combin. Theory,11, 80–84) who proved the above for |W|=n, and also a recent result of B. Bollobás and G. Brightwell (1993,Combinatorica,13, 147–155), ensuring the existence ofC|W|only
Totally Critical Even Order Graphs

AbstractA graph is totally critical if it is Type 2, connected, and the removal of any edge reduces the total chromatic number. A good characterization of all totally critical graphs is unlikely as Sanchez-Arroyo showed that determining the total chromatic number of a graph is an NP-hard problem. In this paper we show that if the Conformability Conjecture is correct, then totally critical graphs of even order with maximum degree at least one half of their order are characterized by a simple equation involving the order, maximum degree, and deficiency of the graph and the edge independence number of the complement. We also show that if the maximum degree Δ of a non-conformable graph G of even order satisfies 12|V|⩽Δ⩽34|V|−1 then G is regular and has a simple structure
Signed Domination in Regular Graphs and Set-Systems

AbstractSuppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a two-coloring of the vertices of G with colors, +1 and −1, such that all closed neighborhoods contain more 1's than −1's, and all together the number of 1's does not exceed the number of −1's by more than (4logr/r+1/r)n. For large r this greatly improves earlier results and is almost optimal, since starting with an Hadamard matrix of order r, a bipartite r-regular graph is constructed on 4r vertices with signed domination number at least (1/2) r−O(1). The determination of limn→∞γs(G)/n remains open and is conjectured to be Θ(1/r)
Norm-Graphs: Variations and Applications

AbstractWe describe several variants of the norm-graphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 12n5/3 edges, containing no copy of K3, 3, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K3, 3 is (1+o(1))k3. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(mt) and whose discrepancy is Ω(n1/2−1/2tlogn). This settles a problem of Matoušek
Stabilizers of Classes of Representable Matroids

AbstractLet M be a class of matroids representable over a field F. A matroid N∈M stabilizes M if, for any 3-connected matroid M∈M, an F-representation of M is uniquely determined by a representation of any one of its N-minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N, or are obtained by a single-element extension followed by a single-element coextension. This result is used to prove that a 3-connected quaternary matroid with no U3, 6-minor has at most (q−2)(q−3) inequivalent representations over the finite field GF(q). New proofs of theorems bounding the number of inequivalent representations of certain classes of matroids are given. The theorem on stabilizers is a consequence of results on 3-connected matroids. It is shown that if N is a 3-connected minor of the 3-connected matroid M, and |E(M)−E(N)|⩾3, then either there is a pair of elements x, y∈E(M) such that the simplifications of M/x, M/y, and M/x, y are all 3-connected with N-minors or the cosimplifications of M\x, M\y, and M\x, y are all 3-connected with N-minors, or it is possible to perform a Δ−Y or Y−Δ exchange to obtain a matroid with one of the above properties
Random Cayley Maps for Groups Generated by Involutions

AbstractLet Γ be a finite group and let Δ be a generating set for Γ. A Cayley map associated with Γ and Δ is an orientable 2-cell imbedding of the Cayley graph GΔ(Γ) such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A probability model for the set of all Cayley maps for a fixed group and generating set, where the distribution is uniform, is investigated in the special case when the generating set contains only elements of order 2. The Cayley genus, maximum Cayley genus, and average Cayley genus are the minimum, maximum, and expected value of the genus random variable associated with this probability model. These parameters are determined for abelian groups, symmetric groups, and dihedral groups (with certain prescribed generating sets). Using the dihedral groups, it is shown that the difference between two consecutive values in the Cayley genus distribution may be arbitrarily large. Further, for each group and its prescribed generating set, the likelihood that a given Cayley map is symmetrical is determined
A Characterization of the Orientations of Ternary Matroids

AbstractA matroid or oriented matroid is dyadic if it has a rational representation with all nonzero subdeterminants in {±2k:k∈Z}. Our main theorem is that an oriented matroid is dyadic if and only if the underlying matroid is ternary. A consequence of our theorem is the recent result of G. Whittle that a rational matroid is dyadic if and only if it is ternary. Along the way, we establish that each whirl has three inequivalent orientations. Furthermore, except for the rank-3 whirl, no pair of these are isomorphic
Weakly Pancyclic Graphs

AbstractA graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. A substantial result of Häggkvist, Faudree, and Schelp (1981) states that a Hamiltonian non-bipartite graph of order n and size at least ⌊(n−1)2/4⌋+2 contains cycles of every length l, 3⩽l⩽n. From this, Brandt (1997) deduced that every non-bipartite graph of the stated order and size is weakly pancyclic. He conjectured the much stronger assertion that it suffices to demand that the size be at least ⌈n2/4⌉−n+5. We almost prove this conjecture by establishing that every graph of order n and size at least ⌊n2/4⌋−n+59 is weakly pancyclic or bipartite
On the Edge Connectivity, Hamiltonicity, and Toughness of Vertex-Transitive Graphs

AbstractLet G be a connected k-regular vertex-transitive graph on n vertices. For S⊆V(G) let d(S) denote the number of edges between S and V(G)\S. We extend results of Mader and Tindell by showing that if d(S)<29(k+1)2 for some S⊆V(G) with 13(k+1)⩽|S|⩽12n, then G has a factor F such that G/E(F) is vertex-transitive and each component of F is an isomorphic vertex-transitive graph on at least 23(k+1) vertices. We show that this result is in some sense best possible and use it to show that if k⩾4 and G has an edge cut of size less than 15(8k−12) which separates G into two components each containing at least two vertices, then G is hamiltonian. We also obtain as a corollary a result on the toughness of vertex-transitive graphs
Clique Minors in Graphs and Their Complements

AbstractA graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t⩾1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c=c(t) independent of G such that the complement of G has a minor isomorphic to Ks, where s=⌈12(1+1/t)n−c⌉. We prove that Kostochka's conjecture is equivalent to the conjecture of Duchet and Meyniel that every graph with no minor isomorphic to Kt+1 has an independent set of size at least n/t. We deduce that Kostochka's conjecture holds for all integers t⩽5, and that a weaker form with s replaced by s′=⌈12(1+1/(2t))n−c⌉ holds for all integers t⩾1
On Random Graph Homomorphisms into Z

AbstractGiven a bipartite connected finite graph G=(V, E) and a vertex v0∈V, we consider a uniform probability measure on the set of graph homomorphisms f:V→Z satisfying f(v0)=0. This measure can be viewed as a G-indexed random walk on Z, generalizing both the usual time-indexed random walk and tree-indexed random walk. Several general inequalities for the G-indexed random walk are derived, including an upper bound on fluctuations implying that the distance d(f(u), f(v)) between f(u) and f(v) is stochastically dominated by the distance to 0 of a simple random walk on Z having run for d(u, v) steps. Various special cases are studied. For instance, when G is an n-level regular tree with all vertices on the last level wired to an additional single vertex, we show that the expected range of the walk is O(logn). This result can also be rephrased as a statement about conditional branching random walk. To prove it, a power-type Pascal triangle is introduced and exploited
Gibbs Measures and Dismantlable Graphs

AbstractWe model physical systems with “hard constraints” by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. Two homomorphisms are deemed to be adjacent if they differ on a single site of G. We investigate what appears to be a fundamental dichotomy of constraint graphs, by giving various characterizations of a class of graphs that we call dismantlable. For instance, H is dismantlable if and only if, for every G, any two homomorphisms from G to H which differ at only finitely many sites are joined by a path in Hom(G, H). If H is dismantlable, then, for any G of bounded degree, there is some assignment of activities to the nodes of H for which there is a unique Gibbs measure on Hom(G, H). On the other hand, if H is not dismantlable (and not too trivial), then there is some r such that, whatever the assignment of activities on H, there are uncountably many Gibbs measures on Hom(Tr, H), where Tr is the (r+1)-regular tree
Matchings in Graphs on Non-orientable Surfaces

AbstractWe generalize Kasteleyn's method of enumerating the perfect matchings in a planar graph to graphs embedding on an arbitrary compact boundaryless 2-manifold S. Kasteleyn stated that perfect matchings in a graph embedding on a surface of genus g could be enumerated as a linear combination of 4g Pfaffians of modified adjacency matrices of the graph. We give an explicit construction that not only does this, but also does it for graphs embedding on non-orientable surfaces. If a graph embeds on the connected sum of a genus g surface with a projective plane (respectively, Klein bottle), the number of perfect matchings can be computed as a linear combination of 22g+1 (respectively, 22g+2) Pfaffians. Thus for any S, this is 22−χ(S) Pfaffians. We also introduce “crossing orientations,” the analogue of Kasteleyn's “admissible orientations” in our context, describing how the Pfaffian of a signed adjacency matrix of a graph gives the sign of each perfect matching according to the number of edge-crossings in the matching. Finally, we count the perfect matchings of an m×n grid on a Möbius strip
Quasi-Hamiltonicity: A Series of Necessary Conditions for a Digraph to Be Hamiltonian

AbstractWe introduce new necessary conditions, k-quasi-hamiltonicity (0⩽k⩽n−1), for a digraph of order n to be hamiltonian. Every (k+1)-quasi-hamiltonian digraph is also k-quasi-hamiltonian; we construct digraphs which are k-quasi-hamiltonian, but not (k+1)-quasi-hamiltonian. We design an algorithm that checks k-quasi-hamiltonicity of a given digraph with n vertices and m arcs in time O(nmk). We prove that (n−1)-quasi-hamiltonicity coincides with hamiltonicity and 1-quasi-hamiltonicity is equivalent to pseudo-hamiltonicity introduced (for undirected graphs) by L. Babel and G. J. Woeginger (1997, in Lecture Notes in Comput. Sci., Vol. 1335, pp. 38–51, Springer-Verlag, New York/Berlin)
Generalized Δ–Y Exchange and k-Regular Matroids

AbstractThis paper introduces a generalization of the matroid operation of Δ–Y exchange. This new operation, segment–cosegment exchange, replaces a coindependent set of k collinear points in a matroid by an independent set of k points that are collinear in the dual of the resulting matroid. The main theorem of the first half of the paper is that, for every field, or indeed partial field, F, the class of matroids representable over F is closed under segment–cosegment exchanges. It follows that, for all prime powers q, the set of excluded minors for GF(q)-representability has at least 2q−4 members. In the second half of the paper, the operation of segment–cosegment exchange is shown to play a fundamental role in an excluded-minor result for k-regular matroids, where such matroids generalize regular matroids and Whittle's near-regular matroids
On Generalized Ramsey Theory: The Bipartite Case

AbstractGiven graphs G and H, a coloring of E(G) is called an (H,q)-coloring if the edges of every copy of H⊆G together receive at least q colors. Let r(G, H, q) denote the minimum number of colors in an (H, q)-coloring of G. We determine, for fixed p, the smallest q for which r(Kn, n, Kp, p, q) is linear in n, the smallest q for which it is quadratic in n. We also determine the smallest q for which r(Kn, n, Kp, p, q)=n2−O(n), and the smallest q for which r(Kn, n, Kp, p, q)=n2−O(1). Our results include showing that r(Kn, n, K2, t+1, 2) and r(Kn, K2, t+1, 2) are both (1+o(1)) n/t as n→∞, thereby proving a special case of a conjecture of Chung and Graham. Finally, we determine the exact value of r(Kn, n, K3, 3, 8), and prove that 2n/3⩽r(Kn, n, C4, 3)⩽n+1. Several problems remain open
Connected Hyperplanes in Binary Matroids

AbstractIn this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently, of Seymour. The following consequence of the main result generalizes a graph result of Thomassen and Toft on induced non-separating cycles and another graph result of Kaugars on deletable vertices. If G is a simple 2-connected graph with minimum degree at least 3, then, for every edge e, there are at least two induced non-separating cycles avoiding e and two deletable vertices non-incident to e. Moreover, G has at least four induced non-separating cycles
2-Symmetric Transformations for 3-Manifolds of Genus 2

AbstractAs previously known, all 3-manifolds of genus 2 can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an elementary transformation on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of genus 2; in fact, it allows an equivalence relation to be defined on admissible 6-tuples so that equivalent 6-tuples represent the same manifold. Different equivalence classes can represent the same manifold; however, equivalence classes “almost always” contain infinitely many 6-tuples. Finally, minimal representatives of the equivalence classes are described
The Speed of Hereditary Properties of Graphs

AbstractGiven a property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. The growth or speed of a property P can be discussed in terms of the values of |Pn|. For properties with |Pn|<nn hereditary properties are surprisingly well determined by their speeds. Sharpening results of E. R. Scheinerman and J. Zito (1994, J. Combin. Theory Ser. B61, 16–39), we prove numerous results about the possible functions |Pn| and describe in detail the properties exhibiting each type of growth. We also list minimal properties exhibiting each type of growth
Least-Distortion Euclidean Embeddings of Graphs: Products of Cycles and Expanders

AbstractEmbeddings of finite metric spaces into Euclidean space have been studied in several contexts: The local theory of Banach spaces, the design of approximation algorithms, and graph theory. The emphasis is usually on embeddings with the least possible distortion. That is, one seeks an embedding that minimizes the bi-Lipschitz constant of the mapping. This question has also been asked for embeddings into other normed spaces. However, when the host space is l2 , more can be said: The problem of finding an optimal embedding into l2 can be formulated as a semi-definite program (and can therefore be solved in polynomial time). So far, this elegant statement of the problem has not been applied to any interesting explicit instances. Here we employ this method and examine two families of graphs: (i) products of cycles, and (ii) constant-degree expander graphs. Our results in (i) extend a 30-year-old result of P. Enflo (1969, Ark. Mat.8, 103–105) on the cube. Our results in (ii) provide an alternative proof to the fact that there are n-point metric spaces whose Euclidean distortion is Ω(logn). Furthermore, we show that metrics in the class (ii) are Ω(logn) far from the class l22, namely, the square of the metrics realizable in l2. This is a well studied class which contains all l1 metrics (and therefore also all l2 metrics). Some of our methods may well apply to more general instances where semidefinite programming is used to estimate Euclidean distortions. Specifically, we develop a method for proving the optimality of an embedding. This idea is useful in those cases where it is possible to guess an optimal embedding
On Extremal Multiflows

AbstractGiven an Eulerian multigraph, a subset T of its vertices, and a collection H of subsets of T, we ask how few edge-disjoint paths can contain maximum (A, T\A)-flows, for all A∈H at once. We answer the question for a certain class of hypergraphs H by presenting a strongly polynomial construction of a minimum set of such paths and a min-max formula for its cardinality. The method consists in reducing the problem to maximizing a b-matching in some graph. The result provides a solution to one interesting class of path packing problems
Contractible Subgraphs in 3-Connected Graphs

AbstractA subgraph H of a 3-connected finite graph G is called contractible if H is connected and G−V(H) is 2-connected. This work is concerned with a conjecture of McCuaig and Ota which states that for any given k there exists an f(k) such that any 3-connected graph on at least f(k) vertices possesses a contractible subgraph on k vertices. We prove this for k⩽4 and consider restrictions to maximal planar graphs, Halin graphs, line graphs of 6-edge-connected graphs, 5-connected graphs of bounded degree, and AT-free graphs