Loopless Graph Research Articles

A sharp bound on the size of a connected matroid

This paper proves that a connected matroid M in which a largest circuit and a largest cocircuit have c and c* elements, respectively, has at most 2 cc* elements. It is also shown that if e is an element of M and ce and c* are the sizes of a largest circuit containing e and a largest cocircuit containing e, then IE(M)I < (Ce-1)(Ce -1)+1. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when ce and c* are replaced by the sizes of a smallest circuit containing e and a smallest cocircuit containing e. Moreover, it follows from the second inequality that if u and v are distinct vertices in a 2-connected loopless graph G, then IE(G)l cannot exceed the product of the length of a longest (u, v)-path and the size of a largest minimal edge-cut separating u from v.

Extremal graphs with prescribed circumference and cocircumference

Let G be a loopless 2-connected graph with circumference c and with largest bond of size c ∗ . In an earlier paper, the author proved that |E(G)|⩽ 1 2 cc ∗ . This paper determines all 2-connected graphs that attain equality in this bound showing that these graphs are certain special series–parallel networks.

A tight lower bound on the maximum genus of 3-edge connected loopless graphs

It is shown that the lower bound on the maximum genus of a 3-edge connected loopless graph is at least one-third of its cycle rank. Moreover, this lower bound is tight. There are infinitely many such graphs attaining the bound.

Fractional Colouring and Hadwiger's Conjecture

Let G be a loopless graph with no K p +1 minor. We prove that the “fractional chromatic number” of G is at most 2 p ; that is, it is possible to assign a rational q ( S )⩾0 to every stable set S ⊆ V ( G ) so that ∑ S ∋ v q ( S )=1 for every vertex v , and ∑ S q ( S )⩽2 p .

Upper embedability of graphs

A connected loopless graph that can be embedded on some (orientable or nonorientable) surface such that the size of each face does not exceed 5 is upper embeddable.

The Four-Colour Theorem

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken's in several respects.

An Upper Bound on the Number of Edges of a 2-Connected Graph

Let G be a loopless 2-connected graph whose largest cycle has c edges, and whose largest bond has c* edges. We show that |E(G)| [les ] ½cc*.

A new proof of the four-colour theorem

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announce another proof, still using a computer, but simpler than Appel and Haken’s in several respects.

On the minors of an incidence matrix and its Smith normal form

Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertex-disjoint odd cycles (i.e., the odd tulgeity) of the graph. The problem of determining this number is shown to be NP-hard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution.

Algebraic graph theory without orientation

Let G be an undirected graph with vertices {v 1,v 2,…,>;v ⋎} and edges { e 1, e 2, …, e ϵ }. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if e j is a link incident with v i , 2 if e j is a loop at v i , and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1's to a − 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from R ϵ to R ⋎ defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MM t , to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component.

Graph algebras and graph varieties

Graph algebras establish a connection between graphs (i.e. binary relations) and universal algebras. A structure theorem of Birkhoff-type is given which characterizes graph varieties, i.e. classes of graphs which can be defined by identities for their corresponding graph algebras: A class of finite directed graphs without multiple edges is a graph variety iff it is closed with respect to finite restricted pointed subproducts and isomorphic copies. Several applications are given, e.g., every loopless finite directed graph is an induced subgraph of a direct power of a graph with three vertices. Graphs with bounded chromatic number or density form graph varieties characterizable by identities of special kind.

The complexity of matching with bonds

A bond structure in a finite, undirected, loopless graph G = ( V, E) is a collection F of nonempty, disjoint subsets of E whose union is E. Matching with bonds (MB) is the problem of finding a maximum-weight matching in G such that either all edges in a bond are selected or none of them. Other matching problems with side constraints have been presented in the literature, we show that they are special cases of (MB). Moreover we prove that (MB) is an NP-hard problem even when the graph G is a cycle and every bond F ∈ F ∈ F has cardinality at most two.

The dimension of graphs with respect to the direct powers of a two-element graph

Every finite loopless undirected graph G is isomorphic to an induced subgraph of a suitable finite direct power of the undirected graph G0 with two adjacent vertices 0,1 and one loop at vertex 1. The least natural number m such that G can be represented in this way is called its G0-dimension. We give some upper and lower bounds of this dimension depending on certain other graph invariants and determine its exact values for some special classes of graphs. Some methods to determine a concrete G0-representation, that is an embedding of G into , are presented. Moreover we show that the problem of determining the G0-dimension of a graph is NP-complete.

Linear arboricity for graphs with multiple edges

AbstractAkiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ(G), the linear arboricity la(G) satisfies ⌈Δ(G)/2⌉ ≦ la(G) ≦ ⌈(Δ(G) + 1)/2⌉. Here it is proved that if G is a loopless graph with maximum degree Δ(G) ≦ k and maximum edge multiplicity μ(G) ≦ k − 2n+1 + 1, where k ≧ 2n−2, then la(G) ≦ k − 2n. It is also conjectured that for any loopless graph G, ⌈Δ(G)/2⌉ ≦ la(G) ≦ ⌈(Δ(G) + μ(G))/2⌉.

Two Theorems on Packings of Graphs

A simple, undirected, loopless graph G of order p and size q is called a ( p, q ) graph. Two graphs G and H of the same order are packable if G can be embedded in the complement H ¯ of H . Two theorems are proved in this paper. Theorem 1 gives a complete characterization of two ( p , p − 1) graphs which are packable. Theorem 2 gives a complete characterization of the pairs { T, G }, where T is a tree of order p and G is a ( p, p ) graph, that can be packed. Theorem 1 generalizes some earlier results of N. Sauer and J. Spencer, D. Burns and S. Schuster, and P. J. Slater, S. K. Teo and H. P. Yap. Theorem 2 extends an earlier result of P. J. Slater, S. K. Teo and H. P. Yap.

Largest bipartite subgraphs in triangle‐free graphs with maximum degree three

AbstractLet G be a triangle‐free, loopless graph with maximum degree three. We display a polynomial algorithm which returns a bipartite subgraph of G containing at least ⅘ of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3‐regular, triangle‐free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.

On shortest cocycle covers of graphs

A cocycle (resp. cycle) cover of a graph G is a family C of cocycles (resp. cycles) of G such that each edge of G belongs to at least one member of C. The length of C is the sum of the cardinalities of its members. While it is known (see [5, 6]) that every bridgeless graph G = ( V, E) has a cycle cover of length not greater than 5 3 |E| , it is shown that there exists no α < 2 such that every loopless graph G = ( V, E) has a cocycle cover with length not greater than α | E|. To do this, cocycle covers of minimal length are determined for the complete graphs, thus solving a problem stated at the end of [6].

On separating systems of graphs

Given a finite loopless graph G (resp. digraph D), let σ( G), ϕ( G) and ψ( D) denote the minimal cardinalities of a completely separating system of G, a separating system of G and a separating system of D, respectively. The main results of this paper are: δ(G) = min m m ⌊m/2⌋ ⩾γ(G) and ϕ(G) = ⌉ log 2 γ(G)⌉ where γ(G) denotes the chromatic number of G. (ii) All the problems of determining σ( G), ϕ( G) and ψ( D) are NP-complete.

Bounds on the number of Eulerian orientations

We show that each loopless 2k-regular undirected graph onn vertices has at least\(\left( {2^{ - k} \left( {_k^{2k} } \right)} \right)^n \) and at most\(\sqrt {\left( {_k^{2k} } \right)^n } \) eulerian orientations, and that, for each fixedk, these ground numbers are best possible.

Solutions to Edmonds' and Katona's problems on families of separating subsets

Let G be a finite loopless graph with vertex-set V( G) and edge-set E( G). Edmonds' problem is to determine the smallest integer, denoted by g( G), for which there exists a family A 1, …, A g( G) of subsets of V( G) such that for any adjacent x i , x j ϵ V( G) there are disjoint A k and A l with x i ϵ A k and x j ϵ A l . Katona's problem is the special case of Edmonds' problem when G is a complete graph. Let γ( G) denote the chromatic number of G. In this paper we establish g(G)=min{2p+3⌈log 3(γ(G)/2 p⌉ | p=0,1,2} , describe a simple method of constructing a required family for Katona's problem with minimal cardinality, and prove that the determination of g( G) is NP-complete.