A characterization of 3-connected graphs containing a given graph

It is shown, as a complement to Tutte's theorem, that for a given 3-connected graph K which is not a wheel, a graph G is 3-connected and has a subgraph contractible to K if and only if G can be obtained from K by a finite sequence of line-additions and 3-point-splittings.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

On matroid connectivity

Three types of matroid connectivity, including Tutte's, are defined and shown to generalize corresponding notions of graph connectivity. A theorem of Tutte on cyclically 3-connected graphs, is generalized to matroids.     

A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

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.     

Soldering and point splitting

Several results concerning the generation of all the n-connected graphs from P_n+1 are presented. Necessary and sufficient conditions for two types of operations, soldering and point splitting, to preserve n-connectivity are given.     

Two characterizations of interchange graphs of complete m-partite graphs

A graph G is m-partite if its points can be partitioned into m subsets V₁,…,V_m such that every line joins a point in V_i with a point in V_j, i ≠ j. A complete m-partite graph contains every line joining V_i with V_j. A complete graph K_p has every pair of its p points adjacent. The nth interchange graph I_n(G) of G is a graph whose points can be identified with the K_n+1's of G such that two points are adjacent whenever the corresponding K_n+1's have a K_n in common.Interchange graphs of complete 2-partite and 3-partite graphs have been characterized, but interchange graphs of complete m-partite graphs for m > 3 do not seem to have been investigated. The main result of this paper is two characterizations of interchange graphs of complete m-partite graphs for m ≥ 2.     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

On Reid's characterization of the ternary matroids

In this paper we prove a stronger version of a result of Ralph Reid characterizing the ternary matroids (i.e., the matroids representable over the field of 3 elements, GF(3)). In particular, we prove that a matroid is ternary if it has no seriesminor of type L_n for n ≥ 5 (n cells and n circuits, each of size n ? 1), and no series-minor of type $\text{L}{}_5{}^1$ (dual of L₅), BII (Fano matroid) or BI (dual of type BII). The proof we give does not assume Reid's theorem. Rather we give a direct proof based on the methods (notably the homotopy theorem) developed by Tutte for proving his characterization of regular matroids. Indeed, the steps involved in our proof closely parallel Tutte's proof, but carrying out these steps now becomes much more complicated.     

Topological minors in line graphs — A proof of Zha’s conjecture

In 1992, Xiaoya Zha conjectured that the line graph of a 3-connected non-planar graph contains a subdivision of K ₅. In this paper we prove this conjecture. This result is the main ingredient of [4] where a complete characterization of all the 4-connected claw-free graphs not containing a subdivision of K ₅ is obtained.     

On the number of Hamiltonian cycles in triangulations

It is proved that if a planar triangulation different from K₃ and K₄ contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ? 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangulated graphs.     

6-Valent analogues of Eberhard’s theorem

It is shown that for every sequence of non-negative integers (p _n|1≦n≠3) satisfying the equation {ie19-1} (respectively, =0) there exists a 6-valent, planar (toroidal, respectively) multi-graph that has preciselyp _n n gonal faces for alln, 1≦n≠3. This extends Eberhard’s theorem that deals, in a similar fashion, with 3-valent, 3-connected planar graphs; the equation involved follows from the famous Euler’s equation.     

Mutation on knots and Whitney’s 2-isomorphism theorem

Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S ² can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term “twisting” and “2-switching” respectively. With the twisting operation, we give a pure geometrical proof of Whitney’s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

A generalization of Tutte's 1-factor theorem to countable graphs

A criterion is proved for a countable graph to possess a perfect matching, in terms of “marriage” in bipartite graphs associated with the graph. In the finite case, this yields Tutte's 1-factor theorem. The criterion is conjectured to be valid for general graphs.     

An algorithmic proof of Tutte's f-factor theorem

Let G be a multigraph on n vertices, possibly with loops. An f-factor is a subgraph of G with degree f_i at the ith vertex for i = 1, 2,…, n. Tutte's f-factor theorem is proved by providing an algorithm that either finds an f-factor or shows that it does not exist and does this in O(n³) operations. Note that the complexity bound is independent of the number of edges of G and the degrees f_i. The algorithm is easily altered to handle the problem of looking for a symmetric integral matrix with given row and column sums by assigning loops degree one. A (g,f)-factor is a subgraph of G with degree d_i at the ith vertex, where g_i ? d_i ? f_i, for i = 1,2,…, n. Lovasz's (g,f)-factor theorem is proved by providing an O(n³) algorithm to either find a (g,f)-factor or show one does not exist.     

On Petersen's graph theorem

In this paper we prove the following: let G be a graph with e_G edges, which is (k ? 1)-edge- connected, and with all valences ?k. Let 1?r?k be an integer, then G contains a spanning subgraph H, so that all valences in H are ?r, with no more than ?re_G?k? edges. The proof is based on a useful extension of Tutte's factor theorem [4,5], due to Lovász [3]. For other extensions of Petersen's theorem, see [6,7,8].