pinecone/core-2020-05-10-deduplication huggingface dataset:
Geodesics in Transitive Graphs

AbstractLetPbe a double ray in an infinite graphX, and letdanddPdenote the distance functions inXand inPrespectively. One callsPageodesicifd(x, y)=dP(x, y), for all verticesxandyinP. We give situations when every edge of a graph belongs to a geodesic or a half-geodesic. Furthermore, we show the existence of geodesics in infinite locally-finite transitive graphs with polynomial growth which are left invariant (set-wise) under “translating” automorphisms. As the main result, we show that an infinite, locally-finite, transitive, 1-ended graph with polynomial growth is planar if and only if the complement of every geodesic has exactly two infinite components
Snarks without Small Cycles

AbstractSnarks are nontrivial cubic graphs whose edges cannot be colored with three colors. Jaeger and Swart conjectured that any snark has girth (the length of the shortest cycle) at most 6. This problem is also known as thegirth conjectureof snarks. The aim of this paper is to give a negative solution of this conjecture and construct snarks with arbitrarily large girths. For instance, if we use known constructions of cubic graphs with large girths, then we can explicitly construct cyclically 5-edge-connected snarks of ordernand with girth at least[formula]
On a Class of Polynomials and Its Relation with the Spectra and Diameters of Graphs

AbstractLetλ1>λ2>…>λdbe points on the real line. For everyk=1, 2, …, d, thek-alternatingpolynomialPkis the polynomial of degreekand norm ‖Pk‖∞=max1⩽l⩽d{|Pk(λl)|}⩽1 that attains maximum absolute value at any pointλ∉[λd, λ1]. Because of this optimality property, these polynomials may be thought of as the discrete version of the Chebychev polynomialsTkand, for particular values of the given points,Pkcoincides in fact with the “shifted”Tk. In general, however, those polynomials seem to bear a much more involved structure than Chebychev ones. Some basic properties of thePkare studied, and it is shown how to compute them in general. The results are then applied to the study of the relationship between the (standard or Laplacian) spectrum of a (not necessarily regular) graph or bipartite graph and its diameter, improving previous results
On Critical Edges in Minimal Imperfect Graphs

AbstractAn edge of a graph is calledcritical, if deleting it the stability number of the graph increases, and a nonedge is calledco-critical, if adding it to the graph the size of the maximum clique increases. We prove in this paper, that the minimal imperfect graphs containing certain configurations of two critical edges and one co-critical nonedge are exactly the odd holes or antiholes. Then we deduce some reformulations of thestrong perfect graph conjectureand prove its validity for some particular cases. Among the consequences we prove that the existence in every minimal imperfect graphGof a maximum cliqueQ, for whichG−Qhas one unique optimal coloration, is equivalent to the strong perfect graph conjecture, as well as the existence of a vertexvinV(G) such that the (uniquely colorable) perfect graphG−vhas a “combinatorially forced” color class. These statements contain earlier results involving more critical edges, of Markossian, Gasparian and Markossian, and those of Bacsó and they also imply that a class of partitionable graphs constructed by Chvátal, Graham, Perold, and Whitesides does not contain counterexamples to the strong perfect graph conjecture
Sudden Emergence of a Giantk-Core in a Random Graph

AbstractThek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdős–Rényi random graphG(n, m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork⩾3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=minλ>0λ/πk(λ) andπk(λ)=P{Poisson(λ)⩾k−1}. In particular,c3≈3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ≈pk(λk)n. Hereλkis the minimum point ofλ/πk(λ) andpk(λk)=P{Poisson(λk)⩾k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one
Hamiltonian Square-Paths

AbstractA hamiltonian square-path (-cycle) is one obtained from a hamiltonian path (cycle) by joining every pair of vertices of distance two in the path (cycle). LetGbe a graph onnvertices with minimum degreeδ(G). Posá and Seymour conjectured that ifδ(G)⩾23n, thenGcontains a hamiltonian square-cycle. We prove that ifδ(G)⩾(2n−1)/3, thenGcontains a hamiltonian square-path. A consequence of this result is a theorem of Aigner and Brandt that confirms the caseΔ(H)=2 of the Bollabás–Eldridge Conjecture: ifGandHare graphs onnvertices and (Δ(G)+1)(Δ(H)+1)⩽n+1, thenGandHcan be packed
Outerplanar Partitions of Planar Graphs

AbstractAnouterplanargraph is one that can be embedded in the plane so that all of the vertices lie on one of the faces. We investigate a conjecture of Chartrand, Geller, and Hedetniemi, that every planar graph can be edge-partitioned into two outerplanar subgraphs. We refute the stronger statement that every planarly embedded graph can be edge-partitioned into two outerplanar subgraphs, one of which is outerplanarly embedded. We give a method that yields outerplanar partitions of certain graphs not covered by previous results. We formulate a conjecture about 4-connected maximal planar graphs that implies the original conjecture. Finally, we verify a weaker form of the conjecture in which outerplanar subgraphs are replaced by subgraphs with no homeomorphs ofK4
Regular Embeddings of Canonical Double Coverings of Graphs

AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG⊗K2, can be described in terms of regular embeddings ofG. This allows us to “lift” the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the “derived” maps by employing those of the “base” maps. We apply these results to determining all orientable regular embeddings of the tensor productsKn⊗K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn⊗K2exist only ifnis a prime powerpl, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes
Spin Models on Triangle-Free Connected Graphs

AbstractSpin models were introduced by V. Jones (Pac. J. Math.137(1989), 311–334) to construct invariants of knots and links. A spin model is defined as a pairS=(X, w) of a fine setXand a functionw:X×X→Csatisfying several axioms. LetΛ=(X, E) be a connected graph with the usual metric ∂:X×X→{0, 1, …, d}, whereddenotes the diameter ofΛ. It is shown that, ifΛhas no 3-cycle, and ifS=(X, t∘∂) is a spin model for a mappingt:{0, 1, …, d}→Csatisfying some conditions (which hold if t is injective), thenΛis an almost bipartite distance-regular graph
Knots and Links in Certain Spatial Complete Graphs

AbstractWe give a spatial representation of the complete graphKnwhich contains exactly[formula]Hopf links consisting of pairs of disjoint 3-cycles, where[formula]is the minimum number of pairs of disjoint 3-cycles forming a non-split link contained in any spatial representation ofKn. Furthermore, we show that the above spatial representation ofKncontains exactly[formula]trefoil knots consisting of 7-cycles, where[formula]is the minimum number of 7-cycles forming a non-trivial knot contained in any spatial representation ofKn
Planar Graphs on Nonplanar Surfaces

AbstractIt is shown that embeddings of planar graphs in arbitrary surfaces other than the 2-sphere have a special structure. It turns out that these embeddings can be described in terms of noncontractible curves in the surface, meeting the graph in at most two points (which may taken to be vertices of the graph). The close connection between the homology group of the surface and the planar graph embeddings is perhaps the most interesting aspect of this study. Some important consequences follow from these results. For example, any two embeddings of a planar graph in the same surface can be obtained from each other by means of simple local reembeddings very similar to Whitney's switchings
Graph Minors XV. Giant Steps

AbstractLetGbe a graph with a subgraphHdrawn with high representativity on a surfaceΣ. When can the drawing ofHbe extended “up to 3-separations” to a drawing ofGinΣif we permit a bounded number (κsay) of “vortices” in the drawing ofG, that is, local areas of non-planarity? (The caseκ=0 was studied in the previous paper of this series.) For instance, if there is a path inGwith ends inH, far apart, and otherwise disjoint fromH, then no such extension exists. We are concerned with the converse; if no extension exists, what can we infer aboutG? It turns out that either there is a path as above, or one of two other obstructions is present
Twice Q-Polynomial Distance-Regular Graphs

AbstractLetΓbe a distance-regular graph of diameter at least three. SupposeΓis Q-polynomial with respect to distinct eigenvaluesθandψofΓ. We calculate thekite numberf2j(x, y, z) from the intersection numbers ofΓand the dual eigenvalues toθandψ. This implies that the kite numbersfij(x, y, z) are independent of the verticesx,y,z. We use this result to show that the vertex neighborhood graphs ofΓare strongly regular and we calculate the parameters of these graphs from the kite numbersf2jand the intersection numbers ofΓ
Locally Pseudo-Distance-Regular Graphs

AbstractThe concept of local pseudo-distance-regularity, introduced in this paper, can be thought of as a natural generalization of distance-regularity for non-regular graphs. Intuitively speaking, such a concept is related to the regularity of graphΓwhen it is seen from a given vertex. The price to be paid for speaking about a kind of distance-regularity in the non-regular case seems to be locality. Thus, we find out that there are no genuine “global” pseudo-distance-regular graphs: when pseudo-distance-regularity is shared by all the vertices, the graph turns out to be distance-regular. Our main result is a characterization of locally pseudo-distance-regular graphs, in terms of the existence of the highest-degree member of a sequence of orthogonal polynomials. As a particular case, we obtain the following new characterization of distance-regular graphs: A graphΓ, with adjacency matrixA, is distance-regular if and only ifΓhas spectrally maximum diameterD, all its vertices have eccentricityD, and the distance matrixADis a polynomial of degreeDinA
Asymptotics of the Chromatic Index for Multigraphs

AbstractFor a multigraphG, letD(G) denote maximum degree and set[formula]We show that the chromatic indexχ′(G) is asymptotically max{D(G),Ggr;(G)}. The latter is, by a theorem of Edmonds (1965), the fractional chromatic index ofG, and the asymptotics established here are part of a conjecture of the author predicting similar agreement of fractional and ordinary chromatic indices for more general hypergraphs. Of particular interest in the present proof is the use of probabilistic ideas and “hard-core” distributions to go from fractional to ordinary (integer) colorings
Cores and Compactness of Infinite Directed Graphs

AbstractIn this paper we define the property of homomorphic compactness for digraphs. We prove that if a digraphHis homomorphically compact thenHhas a core, although the converse does not hold. We also examine a weakened compactness condition and show that when this condition is assumed, compactness is equivalent to containing a core. We use this result to prove that if a digraphHof sizeκis not compact, then there is a digraphGof size at mostκ+such thatHis not compact with respect toG. We then give examples of some sufficient conditions for compactness
The Rank Facets of the Stable Set Polytope for Claw-Free Graphs

AbstractThis paper provides a complete characterization of the rank facets of the stable set polytope STAB(G) associated with a claw-free graphG. In particular, it is shown that a claw-free graphGproduces a rank facet of STAB(G) if and only if it can be obtained by means of two simple lifting procedures from three basic classes of graphs: (i) cliques, (ii) line graphs of minimal 2-connected hypomatchable graphs, and (iii) circulant graphsCω−1αω+1. As a by-product, a characterization of the rank facets of STAB(G) having right-hand side 2 is given
Bridged Graphs Are Cop-Win Graphs: An Algorithmic Proof

AbstractA graph isbridgedif it contains no isometric cycles of length greater than three. Anstee and Farber established that bridged graphs are cop-win graphs. According to Nowakowski and Winkler and Quilliot, a graph is a cop-win graph if and only if its vertices admit a linear orderingv1, v2, …, vnsuch that every vertexvi,i>1, is dominated by some neighbourvj,j<i, i.e., every vertexvk,k<i, adjacent toviis adjacent tovj, too. We present an alternative proof of the result of Anstee and Farber, which allows us to find such an ordering in time linear in the number of edges. Namely, we show that every ordering of the vertices of a bridged graph produced by the breadth-first search is a “cop-win ordering.
Geodetic Rays and Fibers in One-Ended Planar Graphs

AbstractA fiber in an infinite graph is an equivalence class of rays whereby two rays belong to the same fiber whenever each is contained in ann-neighborhood of the other for somen<∞. As this relation is a refinement of end-equivalence, it is of interest when applied to one-ended graphs, in particular to the class Ga, a*of one-ended, 3-connected, planar graphs whose valences and covalences are finite and at leastaand at leasta*, respectively. Any path in a graph in G4, 4that uses at most ⌊12(ρ*(f)−2)⌋ edges of any incident facef(whose covalence isρ*(f)) is shown to be the unique geodetic path joining its end-vertices. From this is deduced that every edge lies on a geodetic double ray, proving a conjecture of Bonnington, Imrich, and Seifter except in the presence of 3-valent vertices or 3-covalent faces. If all valences are at least 4 and all covalences are at least 6, then all Petrie walks are geodetic double rays. Basic questions concerning geodetic fibers (i.e., that contain a geodetic ray) in the graphs in Ga, a*are resolved, namely: (1) how many are there and (2) are they of finite, countable, or uncountabletype, i.e., is every set S of geodetic rays in the fiber that is maximal subject to no two rays in S containing a common subray finite, countable, or uncountable (respectively)? A representative result is that graphs in G4, 6∪G5, 4contain uncountably many geodetic fibers of finite type; furthermore, every geodetic fiber in these graphs contains at most three pairwise-disjoint geodetic rays, revealing an underlying tree-like structure when growth is exponential. In this vein, it is shown that graphs in G4, 5∪G5, 4admit no nonidentity bounded automorphism
Covering Regular Graphs

AbstractA covering projection from a graphGonto a graphHis a “local isomorphism”: a mapping from the vertex set ofGonto the vertex set ofHsuch that, for everyv∈V(G), the neighborhood ofvis mapped bijectively onto the neighborhood (inH) of the image ofv. We investigate two concepts that concern graph covers of regular graphs. The first one is called “multicovers”: we show that for any regular graphHthere exists a graphGthat allows many different covering projections ontoH. Secondly, we considerpartial covers, which require only thatGbe a subgraph of a cover ofH. As an application of our results we show that there are infinitely many rigid regular graphsHfor which theH-cover problem—deciding if a given graphGcoversH—is NP-complete. This resolves an open problem related to the characterization of graphsHfor whichH-COVER is tractable
A Longest Cycle Version of Tutte's Wheels Theorem

AbstractAn edgeeof a minimally 3-connected graphGis non-essential if and only if the graph obtained by contractingefromGis both 3-connected and simple. Suppose thatGis not a wheel. Tutte's Wheels Theorem states thatGhas at least one non-essential edge. We show that each longest cycle ofGcontains at least two non-essential edges. Moreover, each cycle ofGwhose edge set is not contained in a fan contains at least two non-essential edges. We characterize the minimally 3-connected graphs which contain a longest cycle containing exactly two non-essential edges
The Membership Problem in Jump Systems

AbstractA jump system is a set of lattice points satisfying a certain exchange axiom. This notion was introduced by Bouchet and Cunningham [2], as a common generalization of (among others) the sets of bases of a matroid and degree sequences of subgraphs of a graph. We prove, under additional assumptions, a min-max formula for the distance of a lattice point from a jump system. The conditions are met in the examples above, and so our formula contains, as special cases, Tutte'sf-factor-theorem and Edmonds' matroid intersection theorem
Factorization of the Cover Polynomial

AbstractChung and Graham's cover polynomial is a generalization of the factorial rook polynomial in which the second variable keeps track of cycles. We factor the cover polynomial completely for Ferrers boards with either increasing or decreasing column heights. For column-permuted Ferrers boards, we find a sufficient condition for partial factorization. We apply this result to several special cases, including column-permuted “staircase boards,” getting a partial factorization in terms of the column permutation, as well as a sufficient condition for complete factorization. We conclude with some conjectures
Decomposition of Graphs on Surfaces

AbstractLetG=(V, E) be an Eulerian graph embedded on a triangulizable surfaceS. We show thatEcan be decomposed into closed curvesC1, …, Cksuch that mincr(G, D)=∑ki=1mincr(Ci, D) for each closed curveDonS. Here mincr(G, D) denotes the minimum number of intersections ofGandD′ (counting multiplicities), whereD′ ranges over all closed curvesD′ freely homotopic toDand not intersectingV. Moreover, mincr(C, D) denotes the minimum number of intersections ofC′ andD′ (counting multiplicities), whereC′ andD′ range over all closed curves freely homotopic toCandD, respectively.Decomposingthe edges means thatC1, …, Ckare closed curves inGsuch that each edge is traversed exactly once byC1, …, Ck. So each vertexvis traversed exactly 12deg(v) times, where deg(v) is the degree of v. This result was shown by Lins for the projective plane and by Schrijver for compact orientable surfaces. The present paper gives a shorter proof than the one given for compact orientable surfaces. We derive the following fractional packing result for closed curves of given homotopies in a graphG=(V, E) on a compact surfaceS. LetC1, …, Ckbe closed curves onS. Then there exist circulationsf1, …, fk∈REhomotopic toC1, …, Ckrespectively such thatf1(e)+…+fk(e)⩽1 for each edgeeif and only if mincr(G, D)⩾∑ki=1mincr(Ci, D) for each closed curveDonS. Here acirculation homotopicto a closed curveC0is any convex combination of functions trC∈RE, whereCis a closed curve inGfreely homotopic toC0and where trC(e) is the number of timesCtraversese
Making Curves Minimally Crossing by Reidemeister Moves

AbstractLetC1, …, Ckbe a system of closed curves on a triangulizable surfaceS. The system is calledminimally crossingif each curveCihas a minimal number of self-intersections among all curvesC′ifreely homotopic toCiand if each pairCi,Cjhas a minimal number of intersections among all curve pairsC′i, C′jfreely homotopic toCi, Cjrespectively (i, j=1, …, k, i≠j). The system is called regular if each point traversed at least twice by these curves is traversed exactly twice, and forms a crossing. We show that we can make any regular system minimally crossing by applying Reidemeister moves in such a way that at each move the number of crossings does not increase. It implies a finite algorithm to make a given system of curves minimally crossing by Reidemeister moves
The Largest Parity Demigenus of a Simple Graph

AbstractA graphΓisparity embeddedin a surface if a closed path in the graph is orientation preserving or reversing according as its length is even or odd. Theparity demigenusofΓis the minimum of 2−χ(S) (whereχis Euler characteristic) over all surfacesSin whichΓcan be parity embedded. We calculate the maximum parity demigenus over all loopless graphs of ordern. As a corollary we strengthen the calculation by Jungerman, Stahl, and White of the genus ofKn,nbsp;nwith a perfect matching removed. We conclude by discussing numerous related problems
Covering the Edges of a Graph by a Prescribed Tree with Minimum Overlap

AbstractLetH=(VH,EH) be a graph, and letkbe a positive integer. A graphG=(VG,EG) isH-coverable with overlap kif there is a covering of the edges ofGby copies ofHsuch that no edge ofGis covered more thanktimes. Denote by overlap(H,G) the minimumkfor whichGisH-coverable with overlapk. Theredundancyof a covering that usestcopies ofHis (t|EH|−|EG|)/|EG|. Our main result is the following: IfHis a tree onhvertices andGis a graph with minimum degreeδ(G)⩾(2h)10+C, whereCis an absolute constant, then overlap(H,G)⩽2. Furthermore, one can find such a covering with overlap 2 and redundancy at most 1.5/δ(G)0.1. This result is tight in the sense that for every treeHonh⩾4 vertices and for every functionf, the problem of deciding if a graph withδ(G)⩾f(h) has overlap(H,G)=1 is NP-complete
The Number of Dependent Arcs in an Acyclic Orientation

AbstractLetGbe a graph withnnodes,eedges, chromatic numberχ, and girthg. In an acyclic orientation ofG, an arc isdependentif its reversal creates a cycle. It is well known that ifχ<g, thenGhas an acyclic orientation without dependent arcs. Edelman showed that ifGis connected, then every acyclic orientation has at moste−n+1 dependent arcs. We show that ifGis connected andχ<g, thenGhas an acyclic orientation with exactlyddependent arcs for alld⩽e−n+1. We also give bounds on the minimum number of dependent arcs in graphs withχ⩾g
Flows, View Obstructions, and the Lonely Runner

AbstractWe prove the following result: LetGbe an undirected graph. IfGhas a nowhere zero flow with at mostkdifferent values, then it also has one with values from the set {1, …, k}. Whenk⩾5, this is a trivial consequence of Seymour's “six-flow theorem”. Whenk⩽4 our proof is based on a lovely number theoretic problem which we call the “Lonely Runner Conjecture:” Supposekrunners having nonzero constant speeds run laps on a unit-length circular track. Then there is a time at which all runners are at least 1/(k+1) from their common starting point. This conjecture appears to have been formulated by J. Wills (Monatsch. Math.71, 1967) and independently by T. Cusick (Aequationes Math.9, 1973). This conjecture has been verified fork⩽4 by Cusick and Pomerance (J. Number Theory19, 1984) in a complicated argument involving exponential sums and electronic case checking. A major part of this paper is an elementary selfcontained proof of the casek=4 of the Lonely Runner Conjecture