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Register a new userDegree conditions forcing directed cycles
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Caccetta-Haggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kuhn and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, i.e., the smaller of the minimum indegree and the minimum outdegree, that forces a large oriented graph to contain a directed cycle of a given length not divisible by $3$, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by $3$ yet not equal to $3$, hence fully resolve the conjecture of Kelly, Kuhn and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.
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We show that, in almost every $n$-vertex random directed graph process, a copy of every possible $n$-vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself. Furthermore, given an arbitrary $n$-vertex oriented cycle, we determine the sharp threshold for its appearance in the binomial random directed graph. These results confirm, in a strong form, a conjecture of Ferber and Long.
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $ell$-cycles $C_{ell}$: if every vertex $v in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n geq n_0(ell)$ is sufficiently large, then $G$ admits an even rainbow $ell$-cycle $C_{ell}$. This result is best possible whenever $ell otequiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $ell geq 4$, every large $n$-vertex oriented graph $vec{G} = (V, vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $ell$-cycle $vec{C}_{ell}$. Our latter result relates to one of Kelly, Kuhn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2geq8$ and $d_2geq d_1geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where each $V_i$ induces a graph with maximum degree at most $d_i$. Since Steinbergs Conjecture is false, a partition of $V(G)$ into two sets, where one induces an empty graph and the other induces a forest is not guaranteed. Our main theorem is at the intersection of the two aforementioned research directions. We prove that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where $V_1$ induces a forest with maximum degree at most $3$ and $V_2$ induces a forest with maximum degree at most $4$; this is both a relaxation of Steinbergs conjecture and a strengthening of results by Sittitrai and Nakprasit (2019) in a much stronger form.
We prove that there exists a function $f:mathbb{N}rightarrow mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral ErdH{o}s-Posa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.
We prove for all $kgeq 4$ and $1leqell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $mathcal H$ on $n$ vertices to contain a Hamiltonian $ell$-cycle if $k-ell$ divides $n$ and $n$ is sufficiently large. This extends a result of Han and Zhao for $3$-uniform hypegraphs.
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