Showing papers on "Connectivity published in 1979"
TL;DR: This chapter discusses several most important results in integer programming that have been successfully applied to graph theory and then discusses those fields of graph theory where an integer-programming approach has been most effective.
Abstract: Publisher Summary A very large part of combinatorics deals or can be formulated as to deal with optimization problems in discrete structures. Generally, the constraints and the objective function are linear forms of certain variables that are restricted to integers or, mostly, to 0 and 1. Thus, the combinatorial problem is translated to a linear integer-programming problem. The value of such a translation depends on whether it provides new insight or new methods for the solution. This chapter discusses several most important results in integer programming that have been successfully applied to graph theory and then discusses those fields of graph theory where an integer-programming approach has been most effective. The chapter also discusses many graph theoretical results that have a linear programming flavor but no explicit treatment.
45 citations
TL;DR: In this article, constructive combinatorial proofs are given for recurrence formulas which count, respectively, labeled connected graphs and linked diagrams, which give rise to algorithms for selecting at random a labeled connected graph or a linked diagram.
40 citations
TL;DR: A (0,@l)-graph (@l>=2) is a connected graph in which any two vertices have @l common neighbours or none at all, and such a graph is regular when the diameter is at least four.
34 citations
TL;DR: In this paper, the authors define an orientable map as a realization of a finite connected graph G in a topological orientable surface where the complementary domains of G, the faces of the map are topological open discs.
Abstract: An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while. On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S. The choice of P rather than P –l, which corresponds to the other sense of rotation, makes the map “oriented”.
33 citations
TL;DR: It is shown that when H is not Kh+1, there is an h-coloring of H in which a maximum independent set is monochromatic, and this color class may be assumed to be maximum with respect to the condition that its vertices have degree h.
31 citations
TL;DR: The main Theorem of this paper weakens the condition of edge transitivity and is used to show that the connectivity of the graph of the assignment polytope is equal to its degree, thereby proving a conjecture of Balinski and Russakoff.
27 citations
TL;DR: It is shown that in order to obtain the equality κ1(G) = δ(G), it is sufficient that, for each vertex x of minimum degree in G, the vertices in the neighbourhood N(x) of x have sufficiently large degree sum.
Abstract: Let G be a connected graph of order p ≥ 2, with edge-connectivity κ1(G) and minimum degree δ(G). It is shown her ethat in order to obtain the equality κ1(G) = δ(G), it is sufficient that, for each vertex x of minimum degree in G, the vertices in the neighbourhood N(x) of x have sufficiently large degree sum. This result implies a previous result of Chartrand, which required that δ(G) ≥ [p/2].
16 citations
TL;DR: It is shown that for every pair of integers m > n > 1 there is a graph of point connectivity n whose line graph has point connectivity m.
Abstract: Chartrand and Stewart have shown that the line graph of an n-connected graph is itself n-connected. This paper shows that for every pair of integers m > n > 1 there is a graph of point connectivity n whose line graph has point connectivity m. The corresponding question for line connectivity is also resolved.
8 citations
TL;DR: In this article, the authors study finite, valuated graphs that admit a cyclic orientation and give necessary and sufficient conditions for a valuated graph to admit such an orientation concerning the stars and the bonds of the graph.
Abstract: Starting from problem 4 ofK. Wagner [2],H. Fleischner andP. D. Vestergaard [1] introduce the notion of a value-true walk in a finite, connected graph, the edges of which are valuated with nonnegative integers. Their main theorem states that the existence of such a walk is equivalent to the existence of an orientation of the edges with the following property: For every vertex the sum of the valuations of the incoming edges equals the sum of the valuations of the outgoing edges. Let us call such an orientation a cyclic one. In the present paper we study finite, valuated graphs that admit a cyclic orientation. First, we give two necessary conditions for a valuated graphG to admit a cyclic orientation concerning the stars and the bonds ofG, respectively. (The starS (v) of a vertexv is the set of all edges ofG incident withv.) Then, as the main part of the paper we give a characterization of those graphs for which the star- and the bond-condition is sufficient, respectively (for any valuation of the graph). These characterizations are in terms of constructability from trees andK
3, respectively, as well as in terms of forbidden subgraphs.