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تسجيل مستخدم جديدCounting graph orientations with no directed triangles
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Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{lfloor n^2/4rfloor}$ for $n geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{lfloor n^2/4rfloor}$ such orientations.
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Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $kgeq 4$ where $ngeq 5$ or $kgeq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is t
he number of edges of the $n$-vertex $(k-1)$-partite Turan graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also as
ked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with $tinOmega(n^2)$ whose smallest such set has $1.5t-o(n^2)$ arcs.
We count orientations of $G(n,p)$ avoiding certain classes of oriented graphs. In particular, we study $T_r(n,p)$, the number of orientations of the binomial random graph $G(n,p)$ in which every copy of $K_r$ is transitive, and $S_r(n,p)$, the number
of orientations of $G(n,p)$ containing no strongly connected copy of $K_r$. We give the correct order of growth of $log T_r(n,p)$ and $log S_r(n,p)$ up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particula
r, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(frac{1}{4}+o(1))binom{n}{3}$ triangles.
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schroder paths (resp. little Schroder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: begin{align*} r(n,0)&=sumlimits_{j=0}^{
n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+sumlimits_{j=k}^{n-1}2^{j-k}r(n-1,j),quad 1le kle n, s(n,0) &=sumlimits_{j=1}^{n-1}2cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+sumlimits_{j=k+1}^{n-1}2cdot3^{j-k-1}s(n-1,j),quad 1le kle n. end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,kge 0}$ and $[s(n,k)]_{n,kge 0}$, whose row sums produce the large and little Schroder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their $A$- and $Z$-sequences characterizations. On the other hand, it is well-known that the large Schroder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,kge 0}$ as well.
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