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Register a new userNon-crossing Monotone Paths and Binary Trees in Edge-ordered Complete Geometric Graphs
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An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,ldots,k-2$; it is called decreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,ldots,k-2$. We say that $P$ is monotone if it is increasing or decreasing. A rooted tree $T$ in an edge-ordered graph is called monotone if either every path from the root of to a leaf is increasing or every path from the root to a leaf is decreasing. Let $G$ be a graph. In a straight-line drawing $D$ of $G$, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $overline{alpha}(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing path of length $overline{alpha}(G)$. Let $overline{tau}(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing complete binary tree of size $overline{tau}(G)$. In this paper we show that $overline alpha(K_n) = Omega(loglog n)$, $overline alpha(K_n) = O(log n)$, $overline tau(K_n) = Omega(loglog log n)$ and $overline tau(K_n) = O(sqrt{n log n})$.
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An edge-ordering of a graph $G=(V,E)$ is a bijection $phi:Eto{1,2,...,|E|}$. Given an edge-ordering, a sequence of edges $P=e_1,e_2,...,e_k$ is an increasing path if it is a path in $G$ which satisfies $phi(e_i)<phi(e_j)$ for all $i<j$. For a graph $G$, let $f(G)$ be the largest integer $ell$ such that every edge-ordering of $G$ contains an increasing path of length $ell$. The parameter $f(G)$ was first studied for $G=K_n$ and has subsequently been studied for other families of graphs. This paper gives bounds on $f$ for the hypercube and the random graph $G(n,p)$.
In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is used to obtain the explicit formulas for some special cases. Bijection is also established between generalized non-crossing trees with special pattern avoidance and the little Schr{o}der paths.
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of $mathbb{G}(n,omega(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-varepsilon)n$ vertices, where $varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in $mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$. Given an $n$-vertex graph $G$ with minimum degree at least $delta n$, where $delta > 0$ is fixed, we use our aforementioned result in order to prove that a uniform colouring of the randomly perturbed graph $G cup mathbb{G}(n,omega(1)/n)$, using $(1+alpha)n$ colours, where $alpha > 0$ is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree {sl spanning} tree. This can be viewed as a rainbow variant of a result by Krivelevich, Kwan, and Sudakov who proved that $G cup mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$, a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with $G$ as above, we prove that a uniform colouring of $G cup mathbb{G}(n,omega(n^{-2}))$ using $n-1$ colours a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $delta^c(G)geq frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $delta^c(G)geq frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $ngeq 20$, if $delta^c(G)geq frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $ngeq 8$ and $delta^c(G)geq frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $ngeq 20$ to $ngeq 7$, the best possible.
We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.
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