pinecone/core-2020-05-10-deduplication huggingface dataset:
On Tail Distribution of Interpost Distance

AbstractA partial order on Z obtained by taking the transitive closure of a random relation {i<j and there is an edge ij} is studied. Randomness stems from postulating that an edge ij exists with probability p, independently of all other edges. While studying the random order on the subset [n], N. Alon, B. Bollobás, G. Brightwell, and S. Janson (1994, Ann. Appl. Probab.4, 108–123) introduced a remarkable notion of a post, defined as an element in Z comparable to all other elements in the random order. In particular they proved that the interpost distance L has a distribution with a tail Pr(L>x) decreasing at an exponential rate x1/2 at least, whence having all the moments finite. The latter information about L was all they needed in a proof of the central result, asymptotic lognormality of linear extension number. However, it remained unclear whether the exponential rate is actually linear. Our goal in this note is to confirm the conjecture
On the Excluded Minors for Quaternary Matroids

AbstractThis paper strengthens the excluded-minor characterization of GF(4)-representable matroids. In particular, it is shown that there are only finitely many 3-connected matroids that are not GF(4)-representable and that have no U2, 6-, U4, 6-, P6-, F−7-, or (F−7)*-minors. Explicitly, these matroids are all minors of S(5, 6, 12) with rank and corank at least 4, and P″8, the matroid that can be obtained from S(5, 6, 12) by deleting two elements, contracting two elements, and then relaxing the only pair of disjoint circuit-hyperplanes
A Characterization of Graphs with No Cube Minor

AbstractIn this paper it is shown that any 4-connected graph that does not contain a minor isomorphic to the cube is a minor of the line graph of Vn for some n⩾6 or a minor of one of five graphs. Moreover, there exists a unique 5-connected graph on at least 8 vertices with no cube minor and a unique 4-connected graph with a vertex of degree at least 8 with no cube minor. Further, it is shown that any graph with no cube minor is obtained from 4-connected such graphs by 0-, 1-, and 2-summing, and 3-summing over a specified triangles
Duality Theorems for Finite Structures (Characterising Gaps and Good Characterisations)

AbstractWe provide a correspondence between the subjects of duality and density in classes of finite relational structures. The purpose of duality is to characterise the structures C that do not admit a homomorphism into a given target B by the existence of a homomorphism from a structure A into C. Density is the order-theoretic property of containing no covers (or “gaps”). We show that the covers in the skeleton of a category of finite relational models correspond naturally to certain instances of duality statements, and we characterise these covers
A Near Packing of Two Graphs

AbstractLet G1 and G2 be two graphs on n vertices with Δ(G1)=d1 and Δ(G2)=d2. A packing of G1 and G2 is a set L={H1, H2} such that H1≅G1, H2≅G2, and H1 and H2 are edge disjoint subgraphs of Kn. B. Bollobás and S. E. Eldridge (1978, J. Combin. Theory Ser. B25, 105–124) made the following conjecture. If (d1+1)·(d1+1)⩽n+1, then there is a packing of G1 and G2. The degree condition stated in this conjecture could not be weakened, since there exist two such graphs for which (d1+1)·(d2+1)=n+2, and no packing is possible. N. Sauer and J. Spencer (1978, J. Combin. Theory Ser. B25, 295–302) proved that if d1·d2<12n then there is a packing of G1 and G2. In this paper, a generalization and extension of the Sauer–Spencer result is proven. We define a near packing of degree d of two graphs of order n as a generalization of a packing. In a near packing of degree d, the copies of the two graphs may overlap so that the maximum degree of the subgraph defined by the edges common to both copies is d. Thus a near packing of degree 0 is a packing. The result can then be stated as follows. Let a∈R+. If d1·d2⩽an, then there exists a near packing of degree d of G1 and G2 for some d<2a. Furthermore, if (d1+1)·(d2+1)⩽n+1, then the maximum degree d of the subgraph defined by the edges common to both the copy of G1 and the copy of G2 is at most 1 and its size is no larger than n2+1−d1−d2
Vertex Set Partitions Preserving Conservativeness

AbstractLet G be an undirected graph and P={X1, …, Xn} be a partition of V(G). Denote by G/P the graph which has vertex set {X1, …, Xn}, edge set E, and is obtained from G by identifying vertices in each class Xi of the partition P. Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/P, w) is also a conservative graph. We characterize the conservative graphs (G/P, w), where P is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht)
On a Hopf Algebra in Graph Theory

AbstractWe introduce and start the study of a bialgebra of graphs, which we call the 4-bialgebra, and of the dual bialgebra of 4-invariants. The 4-bialgebra is similar to the ring of graphs introduced by W. T. Tutte in 1946, but its structure is more complicated. The roots of the definition are in low dimensional topology, namely, in the recent theory of Vassiliev knot invariants. In particular, 4-invariants of graphs determine Vassiliev invariants of knots. The relation between the two notions is discussed
Colored Homomorphisms of Colored Mixed Graphs

AbstractThe homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic coloring number and oriented chromatic number, have been recently studied. Improving and combining earlier techniques of N. Alon and T. H. Marshall (1998, J. Algebraic Combin.8, 5–13) and A. Raspaud and E. Sopena (1994, Inform. Process. Lett.51, 171–174) we prove here a general result about homomorphisms of colored mixed graphs which implies all these earlier results about planar graphs. We also determine the exact chromatic number of colored mixed trees. For this, we introduce the notion of colored homomorphism for mixed graphs containing both colored arcs and colored edges
Which Crossing Number Is It Anyway?

AbstractA drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed, twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E0 be a subset of its edges such that there is a drawing of G, in which every edge belonging to E0 crosses any other edge an even number of times. Then g can be redrawn so that the elements of E0 are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-crossing number
On Minimally (n, λ)-Connected Graphs

AbstractA graph G is (n, λ)-connected if it satisfies the following conditions: (1) |V(G)|⩾n+1; (2) for any subset S⊆V(G) and any subset L⊆E(G) with λ|S|+|L|<nλ, G−S−L is connected. The (n, λ)-connectivity is a common extension of both the vertex-connectivity and the edge-connectivity. An (n, 1)-connected graph is an n-(vertex)-connected graph, and a (1, λ)-connected graph is a λ-edge-connected graph. An (n, λ)-connected graph G is said to be minimally (n, λ)-connected if for any edge e in E(G), G−e is not (n, λ)-connected. Let G be a minimally (n, λ)-connected graph and let W be the set of its vertices of degree more than nλ. Then we first prove that for any subset W′ of W, the minimum degree of the subgraph of G induced by the vertex set W′ is less than or equal to λ. This result is an extension of a theorem of Mader, which states that the subgraph of a minimally n-connected graph induced by the vertices of degree more than n is a forest. By using our result, we show that if G is a minimally (n, λ)-connected graph, then (1) |E(G)|⩽λ(|V(G)|+n)2/8 for n+1⩽|V(G)|⩽3n−2; (2) |E(G)|⩽nλ(|V(G)|−n) for |V(G)|⩾3n−1. Furthermore, we study the number of vertices of degree nλ in a minimally nλ-connected graph
Minimal Antichains in Well-founded Quasi-orders with an Application to Tournaments

AbstractWe investigate the minimal antichains (in what is essentially Nash-Williams' sense) in a well-founded quasi-order. We prove the following finiteness theorem: If Q is a well-founded quasi-order and k a fixed natural number, then there is a finite set Λk of minimal antichains of Q with the property that for any ideal I of Q obtained by excluding at most k elements of Q, I is well-quasi-ordered if and only if its intersection with each antichain in Λk is finite. When applied in a suitably sharpened form to an algorithmic problem arising in model theory, this yields a strengthening of the main result of [18]
On the Dissection of Rectangles into Right-Angled Isosceles Triangles

AbstractWe consider the problem of dissecting a rectangle or a square into unequal right-angled isosceles triangles. This is regarded as a generalization of the well-known and much-solved problem of dissecting such figures into unequal squares. There is an analogous “electrical” theory but it is based on digraphs instead of graphs and has an appropriate modification of Kirchhoff's first law. The operation of reversing all edges in the digraph is found to be of great help in the construction of “perfect” dissected squares
Contraction–Deletion Invariants for Graphs

AbstractWe consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(G/e)+T(G−e) for e∈E(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc.43, 26–40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z. Extending results of Tutte (1974, J. Combin. Theory Ser. B16, 168–174) and Brylawski (1981, J. Combin. Theory Ser. B30, 233–246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this
Random Matchings Which Induce Hamilton Cycles and Hamiltonian Decompositions of Random Regular Graphs

AbstractSelect four perfect matchings of 2n vertices, independently at random. We find the asymptotic probability that each of the first and second matchings forms a Hamilton cycle with each of the third and fourth. This is generalised to embrace any fixed number of perfect matchings, where a prescribed set of pairs of matchings must each produce Hamilton cycles (with suitable restrictions on the prescribed set of pairs). We also show how the result with four matchings implies that a random d-regular graph for fixed even d⩾4 asymptotically almost surely decomposes into d/2 Hamilton cycles. This completes a general result on the edge-decomposition of a random regular graph into regular spanning subgraphs of given degrees together with Hamilton cycles and verifies conjectures of Janson and of Robinson and Wormald
On Half-Transitive Metacirculant Graphs of Prime-Power Order

AbstractIn this paper, infinitely many graphs of large valency which are half-transitive (that is vertex- and edge-transitive but not arc-transitive) are constructed, and a complete classification is given of half-transitive metacirculant graphs of order a p-power and valency less than 2p, where p is a prime. In particular, it is shown that, for any odd prime p, integers n⩾3 and k⩾2 such that k divides p−1, there are exactly (pn−2−1)/2+pn−3−1 nonisomorphic connected half-transitive metacirculants of order pn and valency 2k
The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

AbstractLet Γ denote a 2-homogeneous bipartite distance-regular graph with diameter D⩾3 and valency k⩾3. Assume that Γ is not isomorphic to a Hamming cube. Fix a vertex x of Γ, and let T=T(x) denote the Terwilliger algebra of T with respect to x. We give three sets of generators for T, two of which satisfy the relations of the quantum universal enveloping algebra of the Lie algebra sl(2). We then describe the simple T-modules. We give a pair of canonical bases for each simple T-module, and we give the overlap function for these bases in terms of a basic hypergeometric function. Finally, we give two generators for the center of T
Nowhere-Zero Flows in Random Graphs

AbstractA nowhere-zero 3-flow in a graph G is an assignment of a direction and a value of 1 or 2 to each edge of G such that, for each vertex v in G, the sum of the values of the edges with tail v equals the sum of the values of the edges with head v. Motivated by results about the region coloring of planar graphs, Tutte conjectured in 1966 that every 4-edge-connected graph has a nowhere-zero 3-flow. This remains open. In this paper we study nowhere-zero flows in random graphs and prove that almost surely as soon as the random graph G(n, p) has minimum degree two it has a nowhere-zero 3-flow. This result is clearly best possible
Determination of all Regular Maps of Small Genus

AbstractComplete lists are given of all reflexible orientable regular maps of genus 2 to 15, all non-orientable regular maps of genus 4 to 30, and all (orientable) rotary but chiral (irreflexible) maps of genus 2 to 15 inclusive. On each list the maps are classified according to genus and type (viz {p, q} where every face is incident with p edges and every vertex is incident with q edges). The complete lists were determined with the help of a parallel program which finds all normal subgroups of low index in a finitely-presented group
High-Girth Graphs Avoiding a Minor are Nearly Bipartite

AbstractLet H be a fixed graph. We show that any H-minor free graph G of high enough girth has circular chromatic number arbitrarily close to two. Equivalently, each such graph G admits a homomorphism to a large odd circuit. In particular, graphs of high girth and of bounded genus, or of bounded tree width, are “nearly bipartite” in this sense. For example, any planar graph of girth at least 16 admits a homomorphism to a pentagon. We also obtain tight bounds on the girth of G in a few specific cases of small forbidden minors H
Drawings of Cm×Cn with One Disjoint Family II

AbstractA long-standing conjecture states that the crossing number of the Cartesian product of cycles Cm×Cn is (m−2)n, for every m, n satisfying n⩾m⩾3. A crossing is proper if it occurs between edges in different principal cycles. In this paper drawings of Cm×Cn with the principal n-cycles pairwise disjoint or the principal m-cycles pairwise disjoint are analyzed, and it is proved that every such drawing has at least (m−2)n proper crossings. As an application of this result, we prove that the crossing number of Cm×Cn is at least (m−2)n/2, for all integers m, n such that n⩾m⩾4. This is the best general lower bound known for the crossing number of Cm×Cn
Anti-Ramsey Colorings in Several Rounds

AbstractA t-round χ-coloring is defined as a sequence ψ1, …, ψt of t (not necessarily distinct) edge colorings of a complete graph, using at most χ colors in each of the colorings. For positive integers k⩽n and t let χt(k, n) denote the minimum number χ of colors for which there exists a t-round χ-coloring of Kn such that all (k2) edges of each Kk⊆Kn get different colors in at least one round. Generalizing a result of J. Körner and G. Simonyi (1995, Studia Sci. Math. Hungar.30, 95–103), it is shown in this paper that χt(3, n)=Θ(n1/t). Two-round colorings for k>3 are also investigated. Tight bounds are obtained for χ2(k, n) for all values of k except for k=5. We also study an inverted extremal function, t(k, n), which is the minimum number of rounds needed to color the edges of Kn with the same (k2) colors such that all (k2) edges of each Kk⊆Kn get different colors in at least one round. For k=n/2, t(k, n) is shown to be exponentially large. Several related questions are investigated. The discussed problems relate to perfect hash functions
A Matrix Method for Chromatic Polynomials

AbstractThe chromatic polynomials of certain families of graphs can be expressed in terms of the eigenspaces of a linear operator. The operator is represented by a matrix, which is referred to here as the compatibility matrix. In this paper complete sets of eigenfunctions are obtained for several related families, and the results are used to provide information about the location of the zeros of the associated chromatic polynomials. A number of uniform features are observed, and these are explained in terms of general properties of the underlying construction
Fragmentability of Graphs

AbstractWe define a parameter which measures the proportion of vertices which must be removed from any graph in a class in order to break the graph up into small (i.e. bounded sized) components. We call this the coefficient of fragmentability of the class. We establish values or bounds for the coefficient for various classes of graphs, particularly graphs of bounded degree. Our main upper bound is proved by establishing an upper bound on the number of vertices which must be removed from a graph of bounded degree in order to leave a planar graph
Line Graphs and Forbidden Induced Subgraphs

AbstractBeineke and Robertson independently characterized line graphs in terms of nine forbidden induced subgraphs. In 1994, Šoltés gave another characterization, which reduces the number of forbidden induced subgraphs to seven, with only five exceptional cases. A graph is said to be a dumbbell if it consists of two complete graphs sharing exactly one common edge. In this paper, we show that a graph with minimum degree at least seven that is not a dumbbell is a line graph if and only if it does not contain three forbidden induced subgraphs including K1, 3 and K5−e. Applications of our main results to other forbidden induced subgraph characterizations of line graphs and to hamiltonian line graphs are also discussed
A Degree Sum Condition for the Existence of a Contractible Edge in a κ-Connected Graph

AbstractIt is known that a noncomplete κ-connected graph of minimum degree of at least ⌊5κ4⌋ contains a κ-contractible edge, i.e., an edge whose contraction yields again a κ-connected graph. Here we prove the stronger statement that a noncomplete κ-connected graph for which the sum of the degrees of any two distinct vertices is at least 2⌊54κ⌋−1 possesses a κ-contractible edge. The bound is sharp and remains valid and sharp if we look only at degree sums at pairs of vertices at distances of one or two, provided that κ≠7
Face Covers and the Genus Problem for Apex Graphs

AbstractA graph G is an apex graph if it contains a vertex w such that G−w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ−1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G−w. The main result of this paper is the linear lower bound g(G)⩾τ/160 (if G−w is 3-connected and τ>1). It is also proved that the minimum face cover problem is NP-hard for planar triangulations and that the minimum vertex cover is NP-hard for 2-connected cubic planar graphs. Finally, it is shown that computing the genus of apex graphs is NP-hard
Matroid 4-Connectivity: A Deletion–Contraction Theorem

AbstractA 3-separation (A, B), in a matroid M, is called sequential if the elements of A can be ordered (a1, …, ak) such that, for i=3, …, k, ({a1, …, ai}, {ai+1, …, ak}∪B) is a 3-separation. A matroid M is sequentially 4-connected if M is 3-connected and, for every 3-separation (A, B) of M, either (A, B) or (B, A) is sequential. We prove that, if M is a sequentially 4-connected matroid that is neither a wheel nor a whirl, then there exists an element x of M such that either M\x or M/x is sequentially 4-connected
Covering Non-uniform Hypergraphs

AbstractA subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)<1.98n(1+o(1)). A special case corresponds to an old problem of Erdős asking for the maximum number of edges in an n-vertex graph with no two cycles of the same length. Denoting this maximum by n+f(n), we can show that f(n)⩽1.98n(1+o(1)). Generalizing the above, let g(n, C, k) denote the maximum of τ(H) taken over all hypergraph H with n vertices and with at most Cik edges with cardinality i for all i=1, 2, …, n. We prove that g(n, C, k)<(Ck!+1)n(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r−1 times. T(n, F, 1)=T(n, F) is the classical Turán number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families
On Structure of Some Plane Graphs with Application to Choosability

AbstractA graph G=(V, E) is (x, y)-choosable for integers x>y⩾1 if for any given family {A(v)∣v∈V} of sets A(v) of cardinality x, there exists a collection {B(v)∣v∈V} of subsets B(v)⊂A(v) of cardinality y such that B(u)∩B(v)=∅ whenever uv∈E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k∈{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable
Graph Imperfection II

AbstractThe imperfection ratio is a graph invariant which indicates how good a lower bound the weighted clique number gives on the weighted chromatic number, in the limit as weights get large. Its introduction was motivated by investigations of the radio channel assignment problem, where one has to assign channels to transmitters and the demands for channels at some transmitters are large. In this paper we show that the imperfection ratio behaves multiplicatively under taking the lexicographic product, which permits us to identify its possible values, investigate its extremal behaviour and its behaviour on random graphs, explore three upper bounds, and show that it is NP-hard to determine