Knot graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta‐wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 100–111, 2000     

Intersection theory for graphs

An intersection theory developed by the author for matroids embedded in uniform geometries is applied to the case when the ambient geometry is the lattice of partitions of a finite set so that the matroid is a graph. General embedding theorems when applied to graphs give new interpretations to such invariants as the dichromate of Tutte. A polynomial in n + 1 variables, the polychromate, is defined for graphs with n vertices. This invariant is shown to be strictly stronger than the dichromate, it is edge-reconstructible and can be calculated for proper graphs from the polychromate of the complementary graph. By using Tutte's construction for codichromatic graphs (J. Combinatorial Theory 16 (1974), 168–174), copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper.     

Novikov homologies in Knot theory

Translated from Matematicheskie Zametki, Vol. 51, No. 3, pp. 53–57, March, 1992.     

Knot theory and the Casson invariant in Artin presentation theory

In Artin presentation theory, a smooth, compact four-manifold is determined by a certain type of presentation of the fundamental group of its boundary. Topological invariants of both three-and four-manifolds can be calculated solely in terms of functions of the discrete Artin presentation. González-Acuña proposed such a formula for the Rokhlin invariant of an integral homology three-sphere. This paper provides a formula for the Casson invariant of rational homology spheres. Thus, all 3D Seiberg-Witten invariants can be calculated by using methods of the theory of groups in Artin presentation theory. The Casson invariant is closely related to canonical knots determined by an Artin presentation. It is also shown that any knot in any three-manifold appears as a canonical knot in Artin presentation theory. An open problem is to determine 4D Seiberg-Witten and Donaldson invariants in Artin presentation theory.     

GROUP THEORY METHODFOR ENUMERATION OFOUTERPLANAR GRAPHS

1.IntroductionAplanargraphiscalledanouterplanargraph[']ifinitsplaneembeddingitsvenicescanbeplacedontheboundaryofaface.Thisfaceisusuallycalledanouterface.Anouterplanargraphissaidtobemaximumifwecannotaddanyedgetokeepitsouterplanarity.Wesupposethatallouterplanargraphsinvestigatedinthispaperaretwoconnected.Theedgesontheboundaryoftheouterfacearecalledouteredgesandotheredgesarecalledinneredgesorchords.Apathu--vconsistedofouteredgeswithd(u)23andd(v)23iscalledasinglechain.Asinglechainissaidtobetrivi…     

Discrete Morse theory on graphs

We characterize the topology of a graph in terms of the critical elements of a discrete Morse function defined on it. Besides, we study the structure and some properties of the gradient vector field induced by a discrete Morse function defined on a graph. Finally, we get results on the number of non-homologically equivalent excellent discrete Morse functions defined on some kind of graphs.     

A polyhedral theory on graphs

This paper provides a polyhedral theory on graphs from which the criteria of Whitney and MacLane for the planarity of graphs are unified, and a brief proof of the Gauss crossing conjecture is obtained. Supported by the National Natural Science Foundation of China.     

A theory of degeneracy graphs

Using some results of the theory of finite sets, a characterization of ×n-degeneracy graphs is developed. This result can be used to derive various structural properties of degeneracy graphs.     

Extremal theory for convex matchings in convex geometric graphs

A convex geometric graphG of ordern consists of the set of vertices of a plane convexn-gonP together with some edges, and/or diagonals ofP as edges. CallG 1-free ifG does not havel disjoint edges in convex position. We answer the following questions:

1. What is the maximum possible number of edges ofG ifG isl-free (as a function ofn andl)?
2. What is the minimum possible number of edges ofG ifG isl-free and saturated, i.e., ifG∪{e} is notl-free for any edge or diagonale ofP that is not, already inG..

We also fully describe the graphsG where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word “disjoint” from the definition of “l-free” and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turán's theorem, etc) in extremal abstract graph theory.     

Online Ramsey theory for a triangle on -free graphs

Given a class of graphs and a fixed graph , the online Ramsey game for H on is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in , and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of forever, and we say Painter wins the game. We initiate the study of characterizing the graphs such that for a given graph , Painter wins the online Ramsey game for on -free graphs. We characterize all graphs such that Painter wins the online Ramsey game for on the class of -free graphs, except when is one particular graph. We also show that Painter wins the online Ramsey game for on the class of -minor-free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead [Electron. J. Combin. 11 (2004), p. 60].     

Julius Petersen''s theory of regular graphs

In 1891 the Danish mathematician Julius Petersen (1839–1910) published a paper on the factorization of regular graphs. This was the first paper in the history of mathematics to contain fundamental results explicitly in graph theory. In this report Petersen's results are analysed and their development in subsequent decades are followed.     

The topological theory of current graphs

W. Gustin's introduction of combinatorial current graphs as a device for obtaining orientable imbeddings of Cayley “color” graphs was fundamental to the solution of the Heawood map-coloring problem by G. Ringel, J. W. T. Youngs, C. M. Terry, and L. R. Welch. The topological current graphs of this paper lead to a construction that generalizes the method of Gustin and its augmentation to “vortex” graphs by Youngs, extending the scope of current graph theory from Cayley graphs alone to the much larger class of graphs that are covering spaces.     

Coloring number and on-line Ramsey theory for graphs and hypergraphs

Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G ₀=(V,E ₀), where E ₀=Ø and V is determined by Builder. On the ith round Builder constructs a new edge e _i (distinct from previous edges) and sets G _i =(V,E _i ), where E _i =E _i?1∪{e _i }. Painter responds by coloring e _i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K _s ^t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges.We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K _s ^t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H ₀ = (V,Ø) with an arbitrary finite number of vertices and no edges. Let H _i?1=(V _i?1,E _i?1) be the hypergraph constructed in the first i ? 1 rounds. On the i-th round Presenter plays by presenting a p-subset P _i ?V _i?1 and Chooser responds by choosing an s-subset X _i ?P _i . The vertices in P _i ? X _i are discarded and the edge X _i added to E _i?1 to form E _i . Presenter wins the survival game if H _i contains a copy of K _s ^t for some i. We show that for positive integers p,s,t with s≤p, Presenter has a winning strategy.     

The theory of guaranteed search on graphs

Pursuit-evasion problems on graphs, the most interesting related results, and many applications of guaranteed search theory in mathematics and other sciences are surveyed.