3‐Flows and Combs

Tutte's 3‐Flow Conjecture states that every 2‐edge‐connected graph with no 3‐cuts admits a 3‐flow. The 3‐Flow Conjecture is equivalent to the following: let G be a 2‐edge‐connected graph, let S be a set of at most three vertices of G; if every 3‐cut of G separates S then G has a 3‐flow. We show that minimum counterexamples to the latter statement are 3‐connected, cyclically 4‐connected, and cyclically 7‐edge‐connected.     

Characterizing 3‐connected planar graphs and graphic matroids

A well‐known result of Tutte states that a 3‐connected graph G is planar if and only if every edge of G is contained in exactly two induced non‐separating circuits. Bixby and Cunningham generalized Tutte's result to binary matroids. We generalize both of these results and give new characterizations of both 3‐connected planar graphs and 3‐connected graphic matroids. Our main result determines when a natural necessary condition for a binary matroid to be graphic is also sufficient. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 165–174, 2010     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

Nowhere‐zero 3‐flows in products of graphs

A graph G is an odd‐circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere‐zero 3‐flow unless G is an odd‐circuit tree and H has a bridge. This theorem is a partial result to the Tutte's 3‐flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere‐zero 3‐flow. A byproduct of this theorem is that every bridgeless Cayley graph G = Cay(Γ,S) on an abelian group Γ with a minimal generating set S admits a nowhere‐zero 3‐flow except for odd prisms. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs

Given a vertex r of a 3-connected graph G, we show how to find three independent spanning trees of G rooted at r. Our proof is based on showing that every 3-connected graph has a nonseparating ear decomposition. This extends Whitney's characterisation that a graph is 2-connected iff it has an ear decomposition. We also show that a nonseparating ear decomposition can be constructed in O(VE) time, and hence, three independent spanning trees can be found in O(VE) time. We construct a nonseparating ear decomposition by solving the following problem at most V times. Given an edge tr and a vertex u of a 3-connected graph G, find a nonseparating induced cycle of G through tr and avoiding u. W. T. Tutte (Proc. London Math. Soc. 13 (1963), 743–767) first showed that such a cycle can always be found. We give a linear time algorithm for this.     

Connectivity and separating sets of cages

A (k; g)-cage is a graph of minimum order among k-regular graphs with girth g. We show that for every cutset S of a (k; g)-cage G, the induced subgraph G[S] has diameter at least ⌊g/2⌋, with equality only when distance ⌊g/2⌋ occurs for at least two pairs of vertices in G[S]. This structural property is used to prove that every (k; g)-cage with k ≥ 3 is 3-connected. This result supports the conjecture of Fu, Huang, and Rodger that every (k; g)-cage is k-connected. A nonseparating g-cycle C in a graph G is a cycle of length g such that G − V(C) is connected. We prove that every (k; g)-cage contains a nonseparating g-cycle. For even g, we prove that every g-cycle in a (k; g)-cage is nonseparating. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 35–44, 1998     

Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs

A graph G is 1‐Hamilton‐connected if G?x is Hamilton‐connected for every x∈V(G), and G is 2‐edge‐Hamilton‐connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X?E⁺(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassen's conjecture (every 4‐connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4‐connected line graph is 1‐Hamilton‐connected and/or 2‐edge‐Hamilton‐connected. As a corollary, we obtain that Thomassen's conjecture implies polynomiality of both 1‐Hamilton‐connectedness and 2‐edge‐Hamilton‐connectedness in line graphs. Consequently, proving that 1‐Hamilton‐connectedness is NP‐complete in line graphs would disprove Thomassen's conjecture, unless P = NP. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012     

On removable circuits in graphs and matroids

Mader proved that every 2‐connected simple graph G with minimum degree d exceeding three has a cycle C, the deletion of whose edges leaves a 2‐connected graph. Jackson extended this by showing that C may be chosen to avoid any nominated edge of G and to have length at least d − 1. This article proves an extension of Jackson's theorem. In addition, a conjecture of Goddyn, van den Heuvel, and McGuinness is disproved when it is shown that a natural matroid dual of Mader's theorem fails. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 51–66, 1999     

Tutte's 5-flow conjecture for graphs of nonorientable genus 5

We develop four constructions for nowhere-zero 5-flows of 3-regular graphs that satisfy special structural conditions. Using these constructions we show a minimal counter-example to Tutte's 5-Flow Conjecture is of order ≥44 and therefore every bridgeless graph of nonorientable genus ≤5 has a nowhere-zero 5-flow. One of the structural properties is formulated in terms of the structure of the multigraph G(F) obtained from a given 3-regular graph G by contracting the cycles of a 2-factor F in G. © 1996 John Wiley & Sons, Inc.     

On removable cycles through every edge

Mader and Jackson independently proved that every 2‐connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G/E(C) is 2‐connected. This paper considers the problem of determining when every edge of a 2‐connected graph G, simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from G remains 2‐connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge‐disjoint removable cycles such that each contains one of the distinguished edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 155–164, 2003     

Fold‐2‐covering triangular embeddings

For a graph G and a positive integer m, G_(m) is the graph obtained from G by replacing every vertex by an independent set of size m and every edge by m² edges joining all possible new pairs of ends. If G triangulates a surface, then it is easy to see from Euler's formula that G_(m) can, in principle, triangulate a surface. For m prime and at least 7, it has previously been shown that in fact G_(m) does triangulate a surface, and in fact does so as a “covering with folds” of the original triangulation. For m = 5, this would be a consequence of Tutte's 5‐Flow Conjecture. In this work, we investigate the case m = 2 and describe simple classes of triangulations G for which G₍₂₎ does have a triangulation that covers G “with folds,” as well as providing a simple infinite class of triangulations G of the sphere for which G₍₂₎ does not triangulate any surface. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 79–92, 2003     

Concept of a vertex in a matroid and 3-connected graphs

The concept of a matroid vertex is introduced. The vertices of a matroid of a 3-connected graph are in one-to-one correspondence with vertices of the graph. Thence directly follows Whitney's theorem that cyclic isomorphism of 3-connected graphs implies isomorphism. The concept of a vertex of a matroid leads to an equally simple proof of Whitney's theorem on the unique embedding of a 3-connected planar graph in the sphere. It also leads to a number of new facts about 3-connected graphs. Thus, consideration of a vertex in a matroid that is the dual of the matroid of a graph leads to a natural concept of a nonseparating cycle of a graph. Whitney's theorem on cyclic isomorphism can be strengthened (even if the nonseparating cycles of a graph are considered, the theorem is found to work) and a new criterion for planarity of 3-connected graphs is obtained (in terms of nonseparating cycles).     

Nowhere‐Zero 5‐Flows On Cubic Graphs with Oddness 4

Tutte's 5‐flow conjecture from 1954 states that every bridgeless graph has a nowhere‐zero 5‐flow. It suffices to prove the conjecture for cyclically 6‐edge‐connected cubic graphs. We prove that every cyclically 6‐edge‐connected cubic graph with oddness at most 4 has a nowhere‐zero 5‐flow. This implies that every minimum counterexample to the 5‐flow conjecture has oddness at least 6.     

On the Thomassen's conjecture*

C. Thomassen proposed a conjecture: Let G be a k‐connected graph with the stability number α ≥ k, then G has a cycle C containing k independent vertices and all their neighbors. In this paper, we will obtain the following result: Let G be a k‐connected graph with stability number α = k + 3 and C any longest cycle of G, then C contains k independent vertices and all their neighbors. This solves Thomassen's conjecture for the case α = k + 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 168–180, 2001     

Upward straight-line embeddings of directed graphs into point sets

In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position.     

Vertex Cuts

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

Short cycle covers of cubic graphs

Let G be a bridgeless cubic graph. We prove that the edges of G can be covered by circuits whose total length is at most (44/27) |E(G)|, and if Tutte's 3-flow Conjecture is true, at most (92/57) |E(G)|.     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory