33 research outputs found
Dirac-type conditions for spanning bounded-degree hypertrees
We prove that for fixed , every -uniform hypergraph on vertices and
of minimum codegree at least contains every spanning tight -tree
of bounded vertex degree as a sub\-graph. This generalises a well-known result
of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is
asymptotically sharp. We also prove an extension of our result to hypergraphs
that satisfy some weak quasirandomness conditions
Clique-Relaxed Graph Coloring
We define a generalization of the chromatic number of a graph G called the k-clique-relaxed chromatic number, denoted χ(k)(G). We prove bounds on χ(k)(G) for all graphs G, including corollaries for outerplanar and planar graphs. We also define the k-clique-relaxed game chromatic number, χg(k)(G), of a graph G. We prove χg(2)(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χg(2)(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χg(2)(H) ≤ 3
Tur\'an Numbers of Ordered Tight Hyperpaths
An ordered hypergraph is a hypergraph whose vertex set is linearly
ordered. We find the Tur\'an numbers for the -uniform -vertex tight path
(with vertices in the natural order) exactly when and
is even; our results imply
when
r\le s}(n,P^{(r)}_s)
remain open. For , we give a construction of an -uniform -vertex
hypergraph not containing which we conjecture to be asymptotically
extremal.Comment: 10 pages, 0 figure