216 research outputs found
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
The domino problem on groups of polynomial growth
We characterize the virtually nilpotent finitely generated groups (or,
equivalently by Gromov's theorem, groups of polynomial growth) for which the
Domino Problem is decidable: These are the virtually free groups, i.e. finite
groups, and those having as a subgroup of finite index
Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex I: An Approximate Asymptotic Result
In this paper and a companion paper, we prove that, if is sufficiently
large, every graph on vertices that has a universal vertex and minimum
degree at least contains each tree with
edges as a subgraph. The present paper already contains an approximate
asymptotic version of the result.
Our result confirms, for large , an important special case of a recent
conjecture by Havet, Reed, Stein, and Wood.Comment: 46 page
Monochromatic cycle partitions in local edge colourings
An edge colouring of a graph is said to be an -local colouring if the
edges incident to any vertex are coloured with at most colours.
Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of
any -locally coloured complete graph may be partitioned into two disjoint
monochromatic cycles of different colours. Moreover, for any natural number
, we show that the vertex set of any -locally coloured complete graph may
be partitioned into disjoint monochromatic cycles. This
generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.Comment: 10 page
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
Spanning Trees in Graphs of High Minimum Degree which have a Universal Vertex II: A Tight Result
We prove that, if is sufficiently large, every graph on vertices
that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result
confirms, for large , an important special case of a conjecture by Havet,
Reed, Stein, and Wood.
The present paper builds on the results of a companion paper in which we
proved the statement for all trees having a vertex that is adjacent to many
leaves.Comment: 29 page
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