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Optimally building spanning graphs in semirandom graph processes

Michael Anastos, Maurício Collares, Joshua Erde, Mihyun Kang, Dominik Schmid, Gregory B. Sorkin

TL;DR

A variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

Abstract

The semirandom graph process constructs a graph $G$ in a series of rounds, starting with the empty graph on $n$ vertices. In each round, a player is offered a vertex $v$ chosen uniformly at random, and chooses an edge on $v$ to add to $G$. The player's aim is to make $G$ satisfy some property as quickly as possible. Our interest is in the property that $G$ contain a given $n$-vertex graph $H$ with maximum degree $Δ$. In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as $n$ tends to infinity, in $(1 + o_Δ(1)) \frac{3 Δn}{2}$ rounds, where $o_Δ(1)$ is a function that tends to $0$ as $Δ$ tends to infinity. We improve this to $(1 + o_Δ(1)) \frac{Δn}{2}$, which can be seen to be asymptotically optimal in $Δ$. We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

Optimally building spanning graphs in semirandom graph processes

TL;DR

A variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on vertices, and chooses one edge to keep.

Abstract

The semirandom graph process constructs a graph in a series of rounds, starting with the empty graph on vertices. In each round, a player is offered a vertex chosen uniformly at random, and chooses an edge on to add to . The player's aim is to make satisfy some property as quickly as possible. Our interest is in the property that contain a given -vertex graph with maximum degree . In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as tends to infinity, in rounds, where is a function that tends to as tends to infinity. We improve this to , which can be seen to be asymptotically optimal in . We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on vertices, and chooses one edge to keep.

Paper Structure

This paper contains 20 sections, 19 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hajnal-Szemerédi Theorem
  4. Chernoff bounds
  5. Concentration of a number of events
  6. Poissonisation
  7. Balanced orientations
  8. The Greedy phase
  9. First subphase: OutGreedy
  10. Second subphase: InGreedy
  11. Repairing failures
  12. Batches and limitations
  13. Workaround
  14. The bridging phase
  15. Final phase
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $H$ be an $n$-vertex graph with maximum degree $\Delta = \Delta(n)$. Then, in the semirandom graph process, Builder has a strategy guaranteeing that whp, after the constructed graph will contain a copy of $H$.

Figures (1)

  • Figure 1: A failed vertex $v$, a swapping candidate $v_1$ and its gadget $R(v,v_1)$ (in blue), and a second swapping candidate $v_2$ and its gadget $R(v,v_2)$ (in green). Vertex $w$ is an illustrative neighbour of $v$; likewise $x$ of $v_1$. All vertices shown distinct are distinct, and the gadgets for $v$ are vertex-disjoint from those for other failed vertices considered at the same time (except that $w$ may be a neighbour of other failed vertices). If all blue edges $R(v,v_1)$ can be made, the $v_1$ gadget succeeds: swapping the labels of $v$ and $v_1$ means both will have edges to their respective $H$-neighbourhoods. Likewise for green and candidate $v_2$. Thin edges will be easy to build. Construction of thick edges is discussed in the text; see particularly \ref{['subsec:batches_workaround']}.

Theorems & Definitions (44)

  • Theorem 1.1: BeGiHeKr2020
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • proof
  • Lemma 2.5: BeGiHeKr2020
  • ...and 34 more