1,441 research outputs found
The Parameterised Complexity of List Problems on Graphs of Bounded Treewidth
We consider the parameterised complexity of several list problems on graphs,
with parameter treewidth or pathwidth. In particular, we show that List Edge
Chromatic Number and List Total Chromatic Number are fixed parameter tractable,
parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even
parameterised by pathwidth. These results resolve two open questions of
Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).Comment: Author final version, to appear in Information and Computation.
Changes from previous version include improved literature references and
restructured proof in Section
Tree-chromatic number is not equal to path-chromatic number
For a graph and a tree-decomposition of , the
chromatic number of is the maximum of , taken
over all bags . The tree-chromatic number of is the
minimum chromatic number of all tree-decompositions of .
The path-chromatic number of is defined analogously. In this paper, we
introduce an operation that always increases the path-chromatic number of a
graph. As an easy corollary of our construction, we obtain an infinite family
of graphs whose path-chromatic number and tree-chromatic number are different.
This settles a question of Seymour. Our results also imply that the
path-chromatic numbers of the Mycielski graphs are unbounded.Comment: 11 pages, 0 figure
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall
The structure of graphs not admitting a fixed immersion
We present an easy structure theorem for graphs which do not admit an
immersion of the complete graph. The theorem motivates the definition of a
variation of tree decompositions based on edge cuts instead of vertex cuts
which we call tree-cut decompositions. We give a definition for the width of
tree-cut decompositions, and using this definition along with the structure
theorem for excluded clique immersions, we prove that every graph either has
bounded tree-cut width or admits an immersion of a large wall
A bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
A linear-time algorithm for the strong chromatic index of Halin graphs
We show that there exists a linear-time algorithm that computes the strong
chromatic index of Halin graphs.Comment: 7 page
Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes
Phil Hanlon proved that the coefficients of the chromatic polynomial of a
graph G are equal (up to sign) to the dimensions of the summands in a
Hodge-type decomposition of the top homology of the coloring complex for G. We
prove a type B analogue of this result for chromatic polynomials of signed
graphs using hyperoctahedral Eulerian idempotents
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