1,279 research outputs found
On DP-Coloring of Digraphs
DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle
and was introduced as an extension of list-colorings of (undirected) graphs. It
transforms the problem of finding a list-coloring of a given graph with a
list-assignment to finding an independent transversal in an auxiliary graph
with vertex set . In this paper, we
extend the definition of DP-colorings to digraphs using the approach from
Neumann-Lara where a coloring of a digraph is a coloring of the vertices such
that the digraph does not contain any monochromatic directed cycle.
Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number,
which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure
Restricted frame graphs and a conjecture of Scott
Scott proved in 1997 that for any tree , every graph with bounded clique
number which does not contain any subdivision of as an induced subgraph has
bounded chromatic number. Scott also conjectured that the same should hold if
is replaced by any graph . Pawlik et al. recently constructed a family
of triangle-free intersection graphs of segments in the plane with unbounded
chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This
shows that Scott's conjecture is false whenever is obtained from a
non-planar graph by subdividing every edge at least once.
It remains interesting to decide which graphs satisfy Scott's conjecture
and which do not. In this paper, we study the construction of Pawlik et al. in
more details to extract more counterexamples to Scott's conjecture. For
example, we show that Scott's conjecture is false for any graph obtained from
by subdividing every edge at least once. We also prove that if is a
2-connected multigraph with no vertex contained in every cycle of , then any
graph obtained from by subdividing every edge at least twice is a
counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our
results to an appendix
Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
The correspondence between on-shell diagrams in maximally supersymmetric
Yang-Mills theory and cluster varieties in the Grassmannian remains largely
unexplored beyond the planar limit. In this article, we describe a systematic
program to survey such 'on-shell varieties', and use this to provide a complete
classification in the case of . In particular, we find exactly 24
top-dimensional varieties and 10 co-dimension one varieties in ---up to
parity and relabeling of the external legs. We use this case to illustrate some
of the novelties found for non-planar varieties relative to the case of
positroids, and describe some of the features that we expect to hold more
generally.Comment: 35 pages, 70 figures, and 1 table; also included is a file with
explicit details for our classification. Signs corrected in two residue
theorems, and a new interpretation (and formula) given for the las
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