Graph reconstruction—a survey

The Reconstruction Conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. This article reviews the progress made on the conjecture since it was first formulated in 1941 and discusses a number of related questions.     

Great-circle tree thrackles

A thrackle is a graph drawing in which every pair of edges meets exactly once. Conway's Thrackle Conjecture states that the number of edges of a thrackle cannot exceed the number of its vertices. Cairns et al. (2015) [1] prove that the Thrackle Conjecture holds for great-circle thrackles drawn on the sphere. They also posit that Conway's Thrackle Conjecture can be restated to say that a graph can be drawn as a thrackle drawing in the plane if and only if it admits a great-circle thrackle drawing. We demonstrate that the class of great-circle thrackleable graphs excludes some trees. Thus the informal conjecture from Cairns et al. (2015) [1] is not equivalent to the Thrackle Conjecture.     

Matching theory and Barnette's conjecture

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity.As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. (1985) [14]. These allow us to check whether a graph we generated is a brace.     

A counterexample to the bold conjecture

A pair of vertices (x,y) of a graph G is an o-critical pair if o(G + xy) > o(G), where G + xy denotes the graph obtained by adding the edge xy to G and o(H) is the clique number of H. The o-critical pairs are never edges in G. A maximal stable set S of G is called a forced color class of G if S meets every o-clique of G, and o-critical pairs within S form a connected graph. In 1993, G. Bacsó raised the following conjecture which implies the famous Strong Perfect Graph Conjecture: If G is a uniquely o-colorable perfect graph, then G has at least one forced color class. This conjecture is called the Bold Conjecture. Here we show a simple counterexample to it. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 165–168, 1997     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n.When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k - 1)m ≤ 2kn, (2) (2k -1)n≤2km, (3) m n ≡ 0 (mod 4k - 1), (4) (4k -1)mn/[2(2k -1)(m n)] is an integer.     

K1,4-自由的模κ泛圈图

设G是2-连通的K1，4自由图．本文证明了当δ(G)≥κ 1时，G是模κ泛圈图．这一结果肯定了猜想2，继而也肯定了Thomassen猜想在2-连通图中的正确性．     

Decomposing planar cubic graphs

The 3‐Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2‐regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

On the Universal Tiling Conjecture in Dimension One

We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints.     

A Contribution to the Second Neighborhood Problem

Seymour’s Second Neighborhood Conjecture asserts that every oriented graph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of oriented graphs whose missing graph is a comb, a complete graph minus two independent edges, or a cycle of length 5.     

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grünbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.     

A semi-strong Perfect Graph theorem

Perfect Graphs were defined by Claude Berge in 1961. Since that time this class of graphs has been intensely studied. Much of the work has been directed towards proving Berge's Strong and Weak Perfect Graph Conjectures. L. Lovász finally demonstrated the Weak Perfect Graph Conjecture in 1972. Vaśek Chvátal, in 1982, proposed the Semi-Strong Perfect Graph Conjecture which falls between these two conjectures. This conjecture suggests that the perfection of a graph depends solely on the way that the chordless paths with three edges are distributed within the graph. This paper contains a proof of Chvátal's conjecture.     

The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. Note that the the Gaussian Moments Conjecture is a special case of [11, Conjecture 3.2]. The latter conjecture was referred to as the Moment Vanishing Conjecture in [7, Conjecture A] and the Integral Conjecture in [6, Conjecture 3.1] (for the one-dimensional case). We also give a counter-example to show that [11, Conjecture 3.2] fails in general for polynomials in more than two variables.     

P<Subscript>4k-1</Subscript>-factorization of complete bipartite graphs

Let K_m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P_v-factorization of K_m,n is a set of edge-disjoint P_v-factors of K_m,n which partition the set of edges of K_m,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of P_v-factorization of K_m,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P_4k-1-factorization of K_m,n is (1) (2k - 1)m ⩽ 2kn, (2) (2k - 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k - 1), (4) (4k - 1)mn/[2(2k - 1)(m + n)] is an integer.     

图全染色的几个定理

图的全染色是染色理论的重要内容 ,全染色猜想 :设 G是一个简单图 ,则 XT( G)≤△ ( G) +2是一个至今未解决的问题 .本文证明了对于一些图类全染色猜想是正确的 .     

Families of graphs complete for the strong perfect graph Conjecture

The Strong Perfect Graph Conjecture states that a graph is perfect iff neither it nor its complement contains an odd chordless cycle of size greater than or equal to 5. In this article it is shown that many families of graphs are complete for this conjecture in the sense that the conjecture is true iff it is true on these restricted families. These appear to be the first results of this type.     

Bidegreed graphs are edge reconstructible

An edge-deleted subgraph of a graph G is a subgraph obtained from G by the deletion of an edge. The Edge Reconstruction Conjecture asserts that every simple finite graph with four or more edges is determined uniquely, up to isomorphism, by its collection of edge-deleted subgraphs. A class of graphs is said to be edge reconstructible if there is no graph in the class with four or more edges that is not edge reconstructible. This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible. The results are then generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructible.     

Decomposition of cubic graphs with a 2-factor consisting of three cycles

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We prove that this conjecture is true for connected cubic graphs with a 2-factor consisting of three cycles.