Rainbow Hamilton cycles in random geometric graphs

Random Structures and Algorithms Pub Date : 2023-11-27 DOI:10.1002/rsa.21201

Alan Frieze, Xavier Pérez-Giménez

引用次数: 1

Abstract

Let

X_{1}, X_{2}, \dots, X_{n} $$ {X}_1,{X}_2,\dots, {X}_n $$

be chosen independently and uniformly at random from the unit

d $$ d $$

-dimensional cube

{[0, 1]}^{d} $$ {\left[0,1\right]}^d $$

. Let

r $$ r $$

be given and let

𝒳 = \{X_{1}, X_{2}, \dots, X_{n}\}

. The random geometric graph

G = G_{𝒳, r}

has vertex set

𝒳

and an edge

X_{i} X_{j} $$ {X}_i{X}_j $$

whenever

‖ X_{i} - X_{j} ‖ \leq r $$ \left\Vert {X}_i-{X}_j\right\Vert \le r $$

. We show that if each edge of

G $$ G $$

is colored independently from one of

n + o (n) $$ n+o(n) $$

colors and

r $$ r $$

has the smallest value such that

G $$ G $$

has minimum degree at least two, then

G $$ G $$

contains a rainbow Hamilton cycle asymptotically almost surely.

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彩虹汉密尔顿循环随机几何图形

设X1 X2…Xn$$ {X}_1,{X}_2,\dots, {X}_n $$ 从单位d中独立均匀随机选取$$ d $$-维立方体[0,1$$ {\left[0,1\right]}^d $$．设r$$ r $$ 设，f =X1,X2，…，Xn。随机几何图G=G∈，r具有顶点集∈∈和一条边∈$$ {X}_i{X}_j $$ $$ \left\Vert {X}_i-{X}_j\right\Vert \le r $$．如果G的每条边$$ G $$ 与n+o(n)中的一个无关$$ n+o(n) $$ 颜色和r$$ r $$ 的值最小，使得G$$ G $$ 最小度至少为2，那么G$$ G $$ 几乎肯定包含一个渐近的彩虹汉密尔顿环。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Structures and Algorithms

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