pinecone/core-2020-05-10-deduplication huggingface dataset:
Dynamic Monopolies of Constant Size

AbstractThe paper deals with a polling game on a graph. Initially, each vertex is colored white or black. At each round, each vertex is colored by the color shared by the majority of vertices in its neighborhood, at the previous round. (All recolorings are done simultaneously.) We say that a set W0 of vertices is a dynamic monopoly or dynamo if starting the game with the vertices of W0 colored white, the entire system is white after a finite number of rounds. D. Peleg (1998, Discrete Appl. Math.86, 262–273) asked how small a dynamic monopoly may be as a function of the number of vertices. We show that the answer is O(1)
A New Bound on the Cyclic Chromatic Number

AbstractIn 1969, Ore and Plummer defined an angular coloring as a natural extension of the Four Color Problem: a face coloring of a plane graph where faces meeting even at a vertex must have distinct colors. A natural lower bound is the maximum degree Δ of the graph. Some graphs require ⌊32Δ⌋ colors in an angular coloring. Ore and Plummer gave an upper bound of 2Δ, which was improved to ⌊95Δ⌋ by the authors with Borodin. This article gives a new upper bound of ⌈59Δ⌉ on the angular chromatic number. The cyclic chromatic number is the equivalent dual vertex coloring problem
Choosability in Random Hypergraphs

AbstractThe choice number of a hypergraph H=(V, E) is the least integer s for which, for every family of color lists S={S(v):v∈V}, satisfying |S(v)|=s for every v∈V, there exists a choice function f so that f(v)∈S(v) for every v∈V, and no edge of H is monochromatic under f. In this paper we consider the asymptotic behavior of the choice number of a random k-uniform hypergraph H(k, n, p). Our main result states that for every k⩾2 and for all values of the edge probability p=p(n) down to p=O(n−k+1) the ratio between the choice number and the chromatic number of H(k, n, p) does not exceed k1/(k−1) asymptotically. Moreover, for large values of p, namely, when p⩾n−(k−1)2/(2k)+ε for an arbitrary positive constant ε, the choice number and the chromatic number of H(k, n, p) have almost surely the same asymptotic value
One or Two Disjoint Circuits Cover Independent Edges Lovász–Woodall Conjecture

AbstractIn this paper, we prove the following theorem: Let L be a set of k independent edges in a k-connected graph G. If k is even or G−L is connected, then there exist one or two disjoint circuits containing all the edges in L. This theorem is the first step in the proof of the conjecture of L. Lovász (1974, Period. Math. Hungar., 82) and D. R. Woodall (1977, J. Combin. Theory Ser. B22, 274–278). In addition, we give the outline of the proof of the conjecture and refer to the forthcoming papers
A Transformation Which Preserves the Clique Number

AbstractWe introduce a graph transformation which preserves the clique number. When applied to graphs containing no odd hole and no cricket (a particular graph on 5 vertices) the transformation also preserves the chromatic number. Using this transformation we derive a polynomial algorithm for the computation of the clique number of all graphs in a class which strictly contains diamond-free graphs. Furthermore, the transformation leads to a proof that the Strong Perfect Graph Conjecture is true for two new classes of graphs and yields a polynomial time algorithm for the computation of the clique number and the chromatic number for both classes. One of these two classes strictly contains claw-free graphs
Patch Colorings and Rigid Colorings of the Rational n-Space

AbstractWe investigate k-colorings of the rational n-space, Qn, such that any two points at distance one get distinct colors. Two types of colorings are considered: patch colorings where the colors occupy open sets with parts of their boundary, and rigid colorings which uniquely extend from any open subset of Qn. We prove that the existence of a patch k-coloring of Qn implies the existence of a k-coloring of Rn. We show that every 2-coloring of Q2 or Q3, and every 4-coloring of Q4 is rigid. We also construct a rigid 3-coloring of Q2
On Essential and Inessential Polygons in Embedded Graphs

AbstractIn this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it has a face chain containing a non-contractible closed path. Our main application is to prove that any two faces of a 4-representative embedding are simultaneously contained in a disc bounded by a polygon. This result is used to prove the existence of ⌊(r−1)/8⌋ pairwise disjoint, pairwise homotopic non-contractible separating polygons in an r -representative orientable embedding. Our proof of this latter result is simple and mechanical
Three Moves on Signed Surface Triangulations

AbstractWe consider finite triangulations of surfaces with signs attached to the faces. Such a signed triangulation is said to have the Heawood property if, at every vertex x, the sum of the signs of the faces incident to x is divisible by 3. For a triangulation G of the sphere, Heawood signings are essentially equivalent to proper 4-vertex-colorings of G. We introduce three moves on signed surface triangulations which preserve the Heawood property. We then prove that every Heawood signed triangulation of the sphere can be obtained from a Heawood signed triangle by a suitable sequence of our moves
Totally Free Expansions of Matroids

AbstractThe aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F -representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r⩾4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour's Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence
Distribution of Cycle Lengths in Graphs

AbstractBondy and Vince proved that every graph with minimum degree at least three contains two cycles whose lengths differ by one or two, which answers a question raised by Erdo&#x030B;s. By a different approach, we show in this paper that if G is a graph with minimum degree δ(G)⩾3k for any positive integer k, then G contains k+1 cycles C0, C1, …, Ck such that k+1<|E(C0)|<|E(C1)|<…<|E(Ck)|, |E(Ci)|− |E(Ci−1)|=2, i⩽i⩽k−1, and 1⩽|E(Ck)|−|E(Ck−1)|⩽2, and furthermore, if δ(G)⩾3k+1, then |E(Ck)|−|E(Ck−1)|=2. To settle a problem proposed by Bondy and Vince, we obtain that if G is a nonbipartite 3-connected graph with minimum degree at least 3k for any positive integer k, then G contains 2k cycles of consecutive lengths m, m+1, …, m+2k−1 for some integer m⩾k+2
Vertex Ramsey Properties of Families of Graphs

AbstractFor graphs F, G1, …, Gr, we write F→(G1, …, Gr) if for every coloring of the vertices of F with r colors there exists i, i=1, 2, …, r, such that a copy of Gi is colored with the ith color. For two families of graphs G1, …, Gr and H1, …, Hs, by (G1, …, Gr)→(H1, …, Hs) we denote the fact that F→(G1, …, Gr) implies F→(H1, …, Hs) for every graph F. In this paper, we give necessary and sufficient conditions for (G1, …, Gr)→(H1, …, Hs) under some weak assumptions on the families of G1, …, Gr and H1, …, Hs. We also consider the induced version of this problem
Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

AbstractWe prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off
Embedding Digraphs on Orientable Surfaces

AbstractWe consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable
Coloring Locally Bipartite Graphs on Surfaces

AbstractIt is proved that there is a function f:N→N such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ⩾f(g). Then (i)If every contractible 4-cycle of G is facial and there is a face of size >4, then G is 3-colorable. (ii)If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C1, …, Cg such that, after cutting along C1, …, Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature
A Continuous Analogue of the Girth Problem

AbstractLet A be the adjacency matrix of a d-regular graph of order n and girth g and d=λ1⩾…⩾λn its eigenvalues. Then ∑nj=2λij=nti−di, for i=0, 1, …, g−1, where ti is the number of closed walks of length i on the d-regular infinite tree. Here we consider distributions on the real line, whose ith moment is also nti−di for all i=0, 1, …, g−1. We investigate distributional analogues of several extremal graph problems involving the parameters n, d, g, and Λ=max|λ2|, |λn|. Surprisingly, perhaps, many similarities hold between the graphical and the distributional situations. Specifically, we show in the case of distributions that the least possible n, given d, g is exactly the (trivial graph-theoretic) Moore bound. We also ask how small Λ can be, given d, g, and n, and improve the best known bound for graphs whose girth exceeds their diameter
Generating Internally Four-Connected Graphs

AbstractA graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. A graph G is internally 4-connected if it is simple, 3-connected, has at least five vertices, and if for every partition (A, B) of the edge-set of G, either |A|⩽3 or |B|⩽3 or at least four vertices of G are incident with an edge in A and an edge in B. We prove that if H and G are internally 4-connected graphs such that they are not isomorphic, H is a minor of G, and they do not belong to a family of exceptional graphs, then there exists a graph H′ such that H′ is isomorphic to a minor of G and either H′ is obtained from H by splitting a vertex or H′ is an internally 4-connected graph obtained from H by means of one of four possible constructions. This is a first step toward a more comprehensive theorem along the same lines
Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice

AbstractThis is a sequel to our papers (M. H. de Carvalho, C. L. Lucchesi, and U. S. R. Murty, 1999, Combinatorica19, 151–174; 2001, J. Combin. Theory, Ser. B; and 2001, J. Combin. Theory, Ser. B). A Petersen brick is a graph whose underlying simple graph is isomorphic to the Petersen graph. For a matching covered graph G,  b(G) denotes the number of bricks of G, and p(G) denotes the number of Petersen bricks of G. An ear decomposition of G is optimal if, among all ear decompositions of G, it uses the least possible number of double ears. Here we make use of the main theorem in (2001, J. Combin. Theory, Ser. B) to prove that the number of double ears in an optimal ear decomposition of a matching covered graph G is b(G)+p(G). In particular, if G is a brick that is not a Petersen brick, then there is an ear decomposition of G with exactly one double ear. This answers a question raised by D. Naddef and W. R. Pulleyblank (1982, Ann. Discrete Math.16, 241–260). Using this theorem, we give an alternative proof of L. Lovász' matching lattice characterization theorem (1987, J. Combin. Theory, Ser. B43, 187–222). We also show that for any matching covered graph G, there is a basis for the matching lattice of G consisting of incidence vectors of perfect matchings of G. This answers a question raised by U. S. R. Murty (1994, “The Matching Lattice and Related Topics,” Technical Report, University of Waterloo). In fact, we show that such a basis may be obtained from the incidence vectors of perfect matchings associated with optimal ear decompositions of G
On a Conjecture of Lovász Concerning Bricks I. The Characteristic of a Matching Covered Graph

AbstractIn 1987, Lovász conjectured that every brick G different from K4, C6, and the Petersen graph has an edge e such that G−e is a matching covered graph with exactly one brick. Lovász and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. In this paper and its sequel (2001, J. Combin. Theory. Ser. B) we present a proof of this conjecture. We shall in fact prove that if G is a brick different from K4, C6, R8 that does not have the Petersen graph as its underlying simple graph, then it has two edges e and f such that both G−e and G−f are matching covered graphs with exactly one brick, with the additional property that, in each case, the underlying simple graph of that one brick is different from the Petersen graph. A cut C of a matching covered graph G is a separating cut if the two C -contractions of G are matching covered. In this paper, we introduce the notion of the characteristic of a separating cut in a matching covered graph and establish some basic properties. We use those properties to first prove our theorem for solid bricks, that is, bricks which do not have any nontrivial separating cuts. The proof of the theorem for nonsolid bricks will be presented in the sequel
Disjoint Cycles in Eulerian Digraphs and the Diameter of Interchange Graphs

AbstractLet R=(r1, …, rm) and S=(s1, …, sn) be nonnegative integral vectors with ∑ri=∑sj. Let A(R, S) denote the set of all m×n {0, 1}-matrices with row sum vector R and column sum vector S. Suppose A(R, S)≠∅. The interchange graphG(R, S) of A(R, S) was defined by Brualdi in 1980. It is the graph with all matrices in A(R, S) as its vertices and two matrices are adjacent provided they differ by an interchange matrix. Brualdi conjectured that the diameter of G(R, S) cannot exceed mn/4. A digraph G=(V, E) is called Eulerian if, for each vertex u∈V, the outdegree and indegree of u are equal. We first prove that any bipartite Eulerian digraph with vertex partition sizes m, n, and with more than (17−1)mn/4 (≈0.78mn) arcs contains a cycle of length at most 4. As an application of this, we show that the diameter of G(R, S) cannot exceed (3+17)mn/16 (≈0.445mn). The latter result improves a recent upper bound on the diameter of G(R, S) by Qian. Finally, we present some open problems concerning the girth and the maximum number of arc-disjoint cycles in an Eulerian digraph
Products of Circulant Graphs Are Metacirculant

AbstractThe standard products—cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by W. Imrich and H. Izbicki (1975, Monatsh. Math.80, 277–281) and later called B-products by I. Broere and J. H. Hattingh (1990, Quaest. Math.13, 191–216) who establish that the lexicographic product of two circulant graphs is again circulant. In this paper, we establish that any B-product of two circulant graphs is always a so-called metacirculant graph with parameters that are easily described in terms of the product graphs. We also establish that any metacirculant graph with the appropriate structure is isomorphic to the B-product of a pair of circulant graphs
Contractible Edges and Triangles in k-Connected Graphs

AbstractIt is proved that if G is a k-connected graph which does not contain K−4, then G has an edge e or a triangle T such that the graph obtained from G by connecting e or by contracting T is still k-connected. By using this theorem, we prove some theorems which are generalizations of earlier work. In addition, we give a condition for a k-connected graph to have a k-contractible edge, which implies two theorems proved by C. Thomassen (1981, J. Graph Theory5, 351–354) and by the author (2001, Australas. J. Combin.24, 165–168), respectively
Proof of a Conjecture of Bollobás and Kohayakawa on the Erdős–Stone Theorem

AbstractFor any integer r⩾1, let a(r) be the largest constant a⩾0 such that if ϵ>0 and 0<c<c0 for some small c0=c0(r, ϵ) then every graph G of sufficiently large order n and at least1−1r+cn2 edges contains a copy of any (r+1)-chromatic graph H of independence numberα(H)⩽(a−ϵ)lognlog(1/c). T. Kővári et al. (1954, Colloq. Math.2, 50–57) and B. Bollobás and P. Erdős (1973, Bull. London Math. Soc.5, 317–321) showed that 1⩽a(1)⩽2. In an improvement to the Erdős–Stone theorem (1946), B. Bollobás et al. (1976, J. London Math. Soc. (2)12, 219–224) showed that a(r)>0 for all r and conjectured that liminfr→∞a(r)≠0. V. Chvátal and E. Szemerédi (1981, J. London Math. Soc. (2)23, 207–214) settled it by giving a(r)⩾0.002 for all r. We show that, for all r,a(r)=a(1). Further we prove the conjecture of B. Bollobás and Y. Kohayakawa (1994, Combinatorica14, 279–286). The weak form of it states that for any r⩾1, 0<c<1/r, every graph G of sufficiently large order n⩾n0(r, c) and (1−1/r+c)(n2) edges contains any (r+1)-chromatic graph such that, in a proper vertex coloring, the smallest and the other color classes are of size at leastβlognlog(1/c) and βlognlogr, respectively, for an absolute constant β>0. That is, all color classes but one are relatively large for fixed r, small c→0, and large n→∞. Our proof method is based on Szemerédi's Regularity Lemma
Families of Regular Graphs in Regular Maps

AbstractThe question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath graphs W(k, n), whose vertices are ordered pairs (i, j), 0⩽i<k, 0⩽j<n, and whose edges are all possible (i, j)−(i+1, j′), and (3) depleted wreath DW(k, n), the subgraph of W(k, n) left by removing all edges in which j=j′. We open the question of multiplicity of occurrence and we list the underlying graphs of rotary maps with no more than 50 edges
Anti-Ramsey Numbers of Subdivided Graphs

AbstractGiven a positive integer n and a family F of graphs, the anti-Ramsey numberf(n, F) is the maximum number of colors in an edge-coloring of Kn such that no subgraph of Kn belonging to F has distinct colors on its edges. The Turán numberex(n, F) is the maximum number of edges of an n-vertex graph that does not contain a member of F as a subgraph. P. Erdős et al. (1975, in Colloq. Math. Soc. Janos Bolyai, Vol. 10, pp. 633–643, North-Holland, Amsterdam) showed for all graphs H that f(n, H)−ex(n, H)=o(n2), where H={H−e:e∈E(H)}. We strengthen their result for the class of graphs in which each edge is incident to a vertex of degree two. We show that f(n, H)−ex(n, H)=O(n) when H belongs to this class. This follows from a new upper bound on f(n, H) that we prove for all graphs H and asymptotically determines f(n, H) for certain graphs H
Coloring Face-Hypergraphs of Graphs on Surfaces

AbstractThe face-hypergraph, H(G), of a graph G embedded in a surface has vertex set V(G), and every face of G corresponds to an edge of H(G) consisting of the vertices incident to the face. We study coloring parameters of these embedded hypergraphs. A hypergraph is k-colorable (k-choosable) if there is a coloring of its vertices from a set of k colors (from every assignment of lists of size k to its vertices) such that no edge is monochromatic. Thus a proper coloring of a face-hypergraph corresponds to a vertex coloring of the underlying graph such that no face is monochromatic. We show that hypergraphs can be extended to face-hypergraphs in a natural way and use tools from topological graph theory, the theory of hypergraphs, and design theory to obtain general bounds for the coloring and choosability problems. To show the sharpness of several bounds, we construct for every even n an n-vertex pseudo-triangulation of a surface such that every triple is a face exactly once. We provide supporting evidence for our conjecture that every plane face-hypergraph is 2-choosable and we pose several open questions, most notably: Can the vertices of a planar graph always be properly colored from lists of size 4, with the restriction on the lists that the colors come in pairs and a color is in a list if and only if its twin color is? An affirmative answer to this question would imply our conjecture, as well as the Four Color Theorem and several open problems
Long Cycles in Graphs on a Fixed Surface

AbstractWe prove that there exists a function a:N0×R+→N such that (i)If G is a 4-connected graph of order n embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1−ε)n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist functions b:N0→N and c:N0→R+ such that for every graph G on n vertices that is embedded on a surface of Euler genus g the following statements hold: (ii)If G is 4-connected, then G contains a collection of at most b(g) paths which cover all vertices of G, and G contains a cycle of length at least n/b(g). (iii)If G is 3-connected, then G contains a cycle of length at least c(g)nlog2/log3. Moreover, for each ε>0, every 4-connected graph G with sufficiently large face-width contains a spanning tree of maximum degree at most 3 and a 2-connected spanning subgraph of maximum degree at most 4 such that the number of vertices of degree 3 or 4 in either of these subgraphs is at most ε|V(G)|