13,125 research outputs found
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Problems and memories
I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the
centennial conference in Budapest to honor Paul Erd\H{o}
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
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