4,722 research outputs found
A note on minimal matching covered graphs
A graph is called matching covered if for its every edge there is a maximum
matching containing it. It is shown that minimal matching covered graphs
contain a perfect matching.Comment: 4 page
A characterization of consistent marked graphs
A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterzing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis
On trees with a maximum proper partial 0-1 coloring containing a maximum matching
I prove that in a tree in which the distance between any two endpoints is
even, there is a maximum proper partial 0-1 coloring such that the edges
colored by 0 form a maximum matching.Comment: 4 page
On Edge-Disjoint Pairs Of Matchings
For a graph G, consider the pairs of edge-disjoint matchings whose union
consists of as many edges as possible. Let H be the largest matching among such
pairs. Let M be a maximum matching of G. We show that 5/4 is a tight upper
bound for |M|/|H|.Comment: 8 pages, 2 figures, Submitted to Discrete Mathematic
The t-improper chromatic number of random graphs
We consider the -improper chromatic number of the Erd{\H o}s-R{\'e}nyi
random graph . The t-improper chromatic number of is
the smallest number of colours needed in a colouring of the vertices in which
each colour class induces a subgraph of maximum degree at most . If ,
then this is the usual notion of proper colouring. When the edge probability
is constant, we provide a detailed description of the asymptotic behaviour
of over the range of choices for the growth of .Comment: 12 page
Quantifying fault recovery in multiprocessor systems
Various aspects of reliable computing are formalized and quantified with emphasis on efficient fault recovery. The mathematical model which proves to be most appropriate is provided by the theory of graphs. New measures for fault recovery are developed and the value of elements of the fault recovery vector are observed to depend not only on the computation graph H and the architecture graph G, but also on the specific location of a fault. In the examples, a hypercube is chosen as a representative of parallel computer architecture, and a pipeline as a typical configuration for program execution. Dependability qualities of such a system is defined with or without a fault. These qualities are determined by the resiliency triple defined by three parameters: multiplicity, robustness, and configurability. Parameters for measuring the recovery effectiveness are also introduced in terms of distance, time, and the number of new, used, and moved nodes and edges
On the Lengths of Symmetry Breaking-Preserving Games on Graphs
Given a graph , we consider a game where two players, and ,
alternatingly color edges of in red and in blue respectively. Let be
the maximum number of moves in which is able to keep the red and the blue
subgraphs isomorphic, if plays optimally to destroy the isomorphism. This
value is a lower bound for the duration of any avoidance game on under the
assumption that plays optimally. We prove that if is a path or a cycle
of odd length , then . The lower
bound is based on relations with Ehrenfeucht games from model theory. We also
consider complete graphs and prove that .Comment: 20 page
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