6,215 research outputs found
Walking Through Waypoints
We initiate the study of a fundamental combinatorial problem: Given a
capacitated graph , find a shortest walk ("route") from a source to a destination that includes all vertices specified by a set
: the \emph{waypoints}. This waypoint routing problem
finds immediate applications in the context of modern networked distributed
systems. Our main contribution is an exact polynomial-time algorithm for graphs
of bounded treewidth. We also show that if the number of waypoints is
logarithmically bounded, exact polynomial-time algorithms exist even for
general graphs. Our two algorithms provide an almost complete characterization
of what can be solved exactly in polynomial-time: we show that more general
problems (e.g., on grid graphs of maximum degree 3, with slightly more
waypoints) are computationally intractable
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
Cycles are strongly Ramsey-unsaturated
We call a graph H Ramsey-unsaturated if there is an edge in the complement of
H such that the Ramsey number r(H) of H does not change upon adding it to H.
This notion was introduced by Balister, Lehel and Schelp who also proved that
cycles (except for ) are Ramsey-unsaturated, and conjectured that,
moreover, one may add any chord without changing the Ramsey number of the cycle
, unless n is even and adding the chord creates an odd cycle.
We prove this conjecture for large cycles by showing a stronger statement: If
a graph H is obtained by adding a linear number of chords to a cycle ,
then , as long as the maximum degree of H is bounded, H is either
bipartite (for even n) or almost bipartite (for odd n), and n is large.
This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses
the regularity method
Decomposing dense bipartite graphs into 4-cycles
Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles
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