arXiv subjects
Publications and source records attributed to Louis DeBiasio.
Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.
A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, Pósa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's theorem by proving that every graph with $deg(u) + deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Châu proved an Ore-type version of Pósa's conjecture for graphs on $n\geq n_0$ vertices using the regularity--blow-up method; consequently the $n_0$ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller $n_0$, we believe that our method of proof will be of independent interest.
In 1960, Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2 (i.e. minimum semi-degree at least n/2), then D contains a directed Hamiltonian cycle. Of course there are other orientations of a cycle in a directed graph and it is not clear that the semi-degree threshold for the directed Hamiltonian cycle is the same as the semi-degree threshold for some other orientation. In 1980, Grant initiated the problem of determining the minimum semi-degree threshold for the anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even n, if D is a directed graph on n vertices with minimum semi-degree at least n/2+1, then D contains an anti-directed Hamiltonian cycle. This result is sharp.
The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladký and Schacht, and Czygrinow and DeBiasio determined the optimal minimum degree condition for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t} for fixed positive integers s<t. For a balanced bipartite graph G[U,V], let δ_U be the minimum degree over all vertices in U and δ_V be the minimum degree over all vertices in V. We consider the problem of determining the optimal value of δ_U+δ_V which guarantees that G can be tiled with K_{s,s}. We show that the optimal value depends on D:=|δ_V-δ_U|. When D is small, we show that δ_U+δ_V\geq n+3s-5 is best possible. As D becomes larger, we show that δ_U+δ_V can be made smaller, but no smaller than n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there exist graphs with δ_U+δ_V\geq n+s^{s^{1/3}} which cannot be tiled with K_{s,s}.
Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.
Given a 3-graph H, let \ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}. Mubayi proved that \ex_2(n,F)=(1/2+o(1))n and conjectured that \ex_2(n, F)=\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Turán density of the family of rings.
Let K_4^3-2e denote the hypergraph consisting of two triples on four points. For an integer n, let t(n, K_4^3-2e) denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree δ_2(G) \geq d contains \floor{n/4} vertex-disjoint copies of K_4^3-2e. Kühn and Osthus proved that t(n, K_4^3-2e) = (1 + o(1))n/4 holds for large integers n. Here, we prove the exact counterpart, that for all sufficiently large integers n divisible by 4, t(n, K_4^3-2e) = n/4 when n/4 is odd, and t(n, K_4^3-2e) = n/4+1 when n/4 is even. A main ingredient in our proof is the recent `absorption technique' of Rödl, Ruciński and Szemerédi.
In 1962 Pósa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 2n/3. Still in 1996, Komlós, Sárközy, and Szemerédi proved Pósa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n > n_0, where n_0 is a very large constant. Here we show without using these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the recent work of Levitt, Szemerédi and Sárközy, but our methods are based on techniques that were available in the 90's.
Bipartite graph tiling was studied by Zhao who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s}. Let s 2s+1. We give the best possible minimum degree condition in this case.