Result Of Tutte Research Articles

Group Connectivity and Group Coloring: Small Groups versus Large Groups

A well-known result of Tutte says that if $\Gamma$ is an Abelian group and $G$ is a graph having a nowhere-zero $\Gamma$-flow, then $G$ has a nowhere-zero $\Gamma'$-flow for each Abelian group $\Gamma'$ whose order is at least the order of $\Gamma$. Jaeger, Linial, Payan, and Tarsi observed that this does not extend to their more general concept of group connectivity. Motivated by this we define $g(k)$ as the least number such that, if $G$ is $\Gamma$-connected for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-connected for every Abelian group $\Gamma'$ of order $|\Gamma'| \geqslant g(k)$. We prove that $g(k)$ exists and satisfies for infinitely many $k$,
 \begin{align*}(2-o(1)) k < g(k) \leqslant 8k^3+1.\end{align*}
 The upper bound holds for all $k$. Analogously, we define $h(k)$ as the least number such that, if $G$ is $\Gamma$-colorable for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-colorable for every Abelian group $\Gamma'$ of order $|\Gamma'| \geq h(k)$. Then $h(k)$ exists and satisfies for infinitely many $k$,
 \begin{align*}(2-o(1)) k < h(k) < (2+o(1))k \ln(k).\end{align*}
 The upper bound (for all $k$) follows from a result of Král', Pangrác, and Voss. The lower bound follows by duality from our lower bound on $g(k)$ as that bound is demonstrated by planar graphs.

On the Complexity Landscape of Connected f-Factor Problems

Let G be an undirected simple graph having n vertices and let $$f:V(G)\rightarrow \{0,\dots , n-1\}$$ be a function. An f-factor of G is a spanning subgraph H such that $$d_H(v)=f(v)$$ for every vertex $$v\in V(G)$$ . The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte (Can J Math 6(1954):347–352, 1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connectedf-factor is easily seen to generalize Hamiltonian Cycle and hence is $$\mathsf {NP}$$ -complete. In fact, the Connected f-Factor problem remains $$\mathsf {NP}$$ -complete even when we restrict f(v) to be at least $$n^{\epsilon }$$ for each vertex v and constant $$0\le \epsilon <1$$ ; on the other side of the spectrum of nontrivial lower bounds on f, the problem is known to be polynomial time solvable when f(v) is at least $$\frac{n}{3}$$ for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restrictions on the function f. In particular, we show that when f(v) is restricted to be at least $$\frac{n}{(\log n)^c}$$ , the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if $$c\le 1$$ . Furthermore, we show that when $$c>1$$ , the problem is $$\mathsf {NP}$$ -intermediate.

Characteristic Flows on Signed Graphs and Short Circuit Covers

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13—28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which occur in two types — either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph $G$ admitting a nowhere-zero $k$-flow has a covering with signed circuits of total length at most $2(k-1)|E(G)|$.

5-Connected Toroidal Graphs are Hamiltonian-Connected

The problem on the Hamiltonicity of graphs is well studied in discrete algorithm and graph theory because of its relation to the traveling salesman problem. Starting with Tutte's result, stating that every 4-connected planar graph is Hamiltonian, several researchers have studied the Hamiltonicity of graphs on surfaces. Extending Tutte's technique, Thomassen proved that every 4-connected planar graph is in fact Hamiltonian-connected, i.e., there is a Hamiltonian path connecting any two prescribed vertices. For graphs on the torus, Thomas and Yu showed that every 5-connected graph on the torus has a Hamiltonian cycle. In this paper, we prove the following result which generalizes Thomas and Yu's result. Every 5-connected graph on the torus is Hamiltonian-connected. Our result is best possible in the sense that we cannot lower the connectivity 5 (i.e., there is a 4-connected graph on the torus which is not Hamiltonian-connected). Moreover, our proof is constructive in a sense that it gives rise to a polynomi...

4-connected projective-planar graphs are Hamiltonian-connected

We generalize the following two seminal results.(1)Thomassen's result [15] in 1983, which says that every 4-connected planar graph is Hamiltonian-connected (which generalizes the old result of Tutte [16] in 1956, which says that every 4-connected planar graph is Hamiltonian).(2)Thomas and Yu's result [12] in 1994, which says that every 4-connected projective-planar graph is Hamiltonian. Here, Hamiltonian-connected means that for any two vertices u,v, there is a Hamiltonian path between u and v (and hence this generalizes the existence of Hamiltonian cycles).Specifically, we prove the following; This proves a conjecture of Dean [3] in 1990. Our result is best possible in many senses. First, we cannot lower the connectivity 4. Secondly, we cannot generalize our result to a surface with higher genus (that is, there is a 4-connected graph on the torus or on the Klein bottle which is not Hamiltonian-connected).Our proof is constructive in a sense that there is a polynomial time algorithm to find, given two vertices in a 4-connected projective-planar graph, a Hamiltonian path between these two vertices.

Characterizing 3‐connected planar graphs and graphic matroids

AbstractA 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

Hamilton Cycles in Planar Locally Finite Graphs

A classical theorem by Tutte ensures the existence of a Hamilton cycle in every finite $4$-connected planar graph. Extensions of this result to infinite graphs require a suitable concept of an infinite cycle. Such a concept was provided by Diestel and Kuhn, who defined circles to be homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. With this definition we prove a partial extension of Tutte's result to locally finite graphs.

Any Maximal Planar Graph with Only One Separating Triangle is Hamiltonian

A graph is hamiltonian if it has a hamiltonian cycle. It is well-known that Tutte proved that any 4-connected planar graph is hamiltonian. It is also well-known that the problem of determining whether a 3-connected planar graph is hamiltonian is NP-complete. In particular, Chvatal and Wigderson had independently shown that the problem of determining whether a maximal planar graph is hamiltonian is NP-complete. A classical theorem of Whitney says that any maximal planar graph with no separating triangles is hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Note that if a planar graph has separating triangles, then it can not be 4-connected and therefore Tutte's result can not be applied. In this paper, we shall prove that any maximal planar graph with only one separating triangle is still hamiltonian.

On random planar graphs, the number of planar graphs and their triangulations

Let P n be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that | P n|⩽n! (75.8) n+o(n) and Bender, Gao and Wormald proved that | P n|⩾n! (26.1) n+o(n) . Gerke and McDiarmid proved that almost all graphs in P n have at least 13/7 n edges. In this paper, we show that | P n|⩽n! (37.3) n+o(n) and that almost all graphs in P n have at most 2.56 n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least ε3 (3 n− m)/2 labelled triangulations on n vertices, where ε is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.

A Characterization of the Matroids Representable over GF(3) and the Rationals

It follows from a fundamental (1958) result of Tutte that a binary matroid is representable over the rationals if and only if it can be represented by a totally unimodular matrix, that is, by a matrix over the rationals with the property that all subdeterminants belong to {0, 1, −1}. For an arbitrary field F , it is of interest to ask for a matrix characterisation of those matroids representable over F and the rationals. In this paper this question is answered when F is GF (3). It is shown that a ternary matroid is representable over the rationals if and only if it can be represented over the rationals by a matrix A with the property that all subdeterminants of A belong to the set {0, ±2 i : i an integer}. While ternary matroids are uniquely representable over GF (3), this is not generally the ease for representations of ternary matroids over other fields. A characterisation is given of the class of ternary matroids that are uniquely representable over the rationals.

A characterization of a class of non-binary matroids

A well-known result of Tutte is that U 2,4, the 4-point line, is the only non-binary matroid M such that, for every element e, both Mβe and M e , the deletion and contraction of e from M, are binary. This paper characterizes those non-binary matroids M such that, for every element e, Mβe or M e is binary.

Minor characterization of undirected branching greedoids—a short proof

In 1959 Tutte [5] gave a minor characterization of graphic matroids. Within the framework of greedoids, a natural analogue of the cycle matroid in graphs is the branching greedoid. Schmidt has shown that, similar to Tutte's result, branching greedoids can be characterized by forbidden minors. Here we give a simpler proof of this theorem.

Forcing constructions for uncountably chromatic graphs

In this paper we solve some of Pál Erdős's favorite problems on uncountably chromatic graphs. Generalizing a finite graph theory result of Tutte, Erdős and R. Rado showed that for every infinite cardinal κ there exists a triangle-free, κ-chromatic graph of size κ. For κ = ℵ0, Erdős established the existence of ℵ0-chromatic graphs excluding even C4, C5,…, Cn, i.e. circuits up to a given length. For κ &lt; ℵ0 the situation is different. As shown by Erdős and A. Hajnal, a graph is necessarily countably chromatic if it omits any finite bipartite graph. We can, however, exclude any finite list of nonbipartite graphs (this obviously reduces to excluding finitely many odd circuits). They posed an even stronger conjecture, namely, that similar examples must occur in every uncountably chromatic graph. To be specific, they conjectured that for every infinite κ, every κ-chromatic graph contains a κ-chromatic triangle-free subgraph. Here we show that this may not be true for κ = ℵ1 i.e. we exhibit a model where it is false. We must emphasize that the conjecture is probably false already in ZFC, but we have been unable to show this.

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.

On minor-minimally-connected matroids

By a well-known result of Tutte, if e is an element of a connected matroid M, then either the deletion or the contraction of e from M is connected. If, for every element of M, exactly one of these minors is connected, then we call M minor-minimally-connected. This paper characterizes such matroids and shows that they must contain a number of two-element circuits or cocircuits. In addition, a new bound is proved on the number of 2-cocircuits in a minimally connected matroid.

The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.

Enumeration of coloured plane trees with a given type partition

Tutte's result for the number of planted plane trees with a given degree partition is rederived by a variety of methods and in particular by a simple piecewise construction technique. A theorem of Gordon and Temple is applied in order to give a general relationship between the number of planted plane trees and the number of rooted plane trees and the degree partition restriction is generalised to type partition. The piecewise construction method is successfully used to derive the number of planted plane trees with a given 2-colour degree partition, also derived by Tutte, and an algorithm for the k-coloured case is developed. This algorithm may be used to obtain more specific results. These models are relevant to the statistical mechanics of polymers and this is discussed briefly.

Matchings in graphs

Results of Tutte and of Anderson giving conditions for a simple graph G to have a perfect matching are generalized to give conditions for G to have a matching of defect d. A corollary to one of these results is a theorem of Berge on the size of a maximum matching in G.

Matchings in Arbitrary Graphs

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

A combinatorial proof of a theorem of Tutte

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) &gt; n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.