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Constructing All Birthday 3 Games as Digraphs

Alexander Clow, Alfie Davies, Neil Anderson McKay

TL;DR

This work determines the minimal vertex count needed to realise all Day-3 digraph placement values, proving $F(3)=8$ and thus that every Day-3 value has a representation on at most $8$ vertices. The authors blend exhaustive enumeration of digraphs on up to $7$ vertices with large-scale random searches on $8$-vertex graphs to certify all Day-3 values, and supply a combinatorial lower bound via a symmetric Day-3 value $\mathcal{Z}$ to establish $f(\mathcal{Z})\ge 8$. They also derive improved bounds for higher birthdays, showing $F(4)\in[11,2087]$ and $F(5)\le 10^{187.63}$ (with a substantial lower bound $F(5)\ge 2.58\times 10^{12}$), by leveraging anti-chain bounds and $D(n)$-based counts of coloured digraphs. Together, these results deepen understanding of the vertex-complexity needed to realize all game values born by days $4$ and $5$ and motivate future work on universal digraph-placement representations and the structure of Day-3 values.

Abstract

Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games $X$, there is a Digraph Placement game $G$ with $G=X$. Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5.

Constructing All Birthday 3 Games as Digraphs

TL;DR

This work determines the minimal vertex count needed to realise all Day-3 digraph placement values, proving and thus that every Day-3 value has a representation on at most vertices. The authors blend exhaustive enumeration of digraphs on up to vertices with large-scale random searches on -vertex graphs to certify all Day-3 values, and supply a combinatorial lower bound via a symmetric Day-3 value to establish . They also derive improved bounds for higher birthdays, showing and (with a substantial lower bound ), by leveraging anti-chain bounds and -based counts of coloured digraphs. Together, these results deepen understanding of the vertex-complexity needed to realize all game values born by days and and motivate future work on universal digraph-placement representations and the structure of Day-3 values.

Abstract

Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games , there is a Digraph Placement game with . Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5.
Paper Structure (5 sections, 15 theorems, 15 equations, 2 figures, 3 tables)

This paper contains 5 sections, 15 theorems, 15 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Constructing all Day 3 Games
  3. Some Day 3 Games Require 8 Vertices
  4. Games Born Day 4 and Day 5
  5. Final remarks

Key Result

Theorem 1.1

For all games $X$ born by day 3, there exists a digraph placement game $G$ with at most 8 vertices such that $G=X$. Furthermore, there exists a game $X$ with birthday 3 such that $H\neq X$ for all digraph placement games $H$ with at most 7 vertices.

Figures (2)

  • Figure 1: A digraph placement game equal to $\mathord\uparrow$ (left) and another equal to $\{ 2 \mathrel\vert -2 \}$ (right). Blue vertices are circles, red vertices are squares, and pairs of arcs $(u,v),(v,u)$ are shown with a bolded edge.
  • Figure 2: An $8$-vertex digraph placement game equal to $\mathop{\mathrm{\mathcal{Z}}}\nolimits$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 16 more