Consistency in marked digraphs

In this paper we strive towards a mathematical theory for "marked digraphs" in which the nodes are signed. For completeness, we begin with an extensive list of definitions, including that of "consistency" in marked digraphs. We then provide three different descriptions of the concept: one an alternative in terms of directed cycles, another in terms of partitioning the nodes, and a third in terms of arc-digraphs and balance. We conclude with two additional observations, one characterizing the structure of consistent strongly connected marked tournaments and the other giving a criterion for a digraph to be "markable" in a consistent way.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22468/1/0000009.pd

The connectivity function of a graph

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/153221/1/mtks0025579300003806.pd

The decycling number of graphs

For a graph G and S ⊂ V (G), if G − S is acyclic, then S is said to be a decycling set of G. The size of a smallest decycling set of G is called the decycling number of G. The purpose of this paper is to provide a review of recent results and open problems on this parameter. Results to be reviewed include recent work on decycling numbers of cubes, grids and snakes and bounds on the decycling number of cubic graphs, and expected bounds on the decycling numbers of random regular graphs. A structural description of graphs with a fixed decycling number based on connectivity is also presented

Graph connections : relationships between graph theory and other areas of mathematics

xi, 291 p. : ill. ; 21x30 cm

Topics in Structural Graph Theory

The rapidly expanding area of structural graph theory uses ideas of connectivity to explore various aspects of graph theory, and vice versa. It has links with other areas of mathematics, and is increasingly used in such areas as computer networks where connectivity algorithms are an important feature. The book begins with an introductory chapter by the editors, followed by thirteen expository chapters, each written by acknowledged experts, It is the third in a series having these Editors, the first two being Topics in Algebraic Graph Theory and Topics in Topological Graph Theory

On the edge-chromatic number of a graph

AbstractV.G. Vizing has shown that the edge-chromatic number of any graph with maximum vertex-degree ρ is equal to either ρ or ρ + 1. In this paper, we describe various ways of constructing graphs whose edge-chromatic number is ρ + 1 and formulate a conjecture about such graphs