4,477 research outputs found
Cycle packing
In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every
graph on n vertices can be partitioned into O(n) cycles and edges. They
observed that one can easily get an O(n log n) upper bound by repeatedly
removing the edges of the longest cycle. We make the first progress on this
problem, showing that O(n log log n) cycles and edges suffice. We also prove
the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear
minimum degree.Comment: 18 page
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Cycle lengths in sparse graphs
Let C(G) denote the set of lengths of cycles in a graph G. In the first part
of this paper, we study the minimum possible value of |C(G)| over all graphs G
of average degree d and girth g. Erdos conjectured that |C(G)|
=\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this
conjecture. In particular, the longest cycle in a graph of average degree d and
girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this
problem was initiated by Ore in 1967 and our result improves all previously
known lower bounds on the length of the longest cycle. Moreover, our bound
cannot be improved in general, since known constructions of d-regular Moore
Graphs of girth g have roughly that many vertices. We also show that
\Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle
lengths in a graph of chromatic number d and girth g. Further results are
obtained for the number of cycle lengths in H-free graphs of average degree d.
In the second part of the paper, motivated by the conjecture of Erdos and
Gyarfas that every graph of minimum degree at least three contains a cycle of
length a power of two, we prove a general theorem which gives an upper bound on
the average degree of an n-vertex graph with no cycle of even length in a
prescribed infinite sequence of integers. For many sequences, including the
powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average
degree of graph of order n with no cycle of length in the sequence, where
\log^* n is the number of times the binary logarithm must be applied to n to
get a number which is at mos
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