COMBINATORICA· 2026Q1
Making Spanning Graphs
- 4citations
- Q1SCImago
- 2026year
Short summary
Maker can guarantee claiming a graph G containing all graphs H with up to n vertices and maximum degree D, provided the bias b is at most c(n/log n)^(1/D).
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Key points
- Maker-Breaker game on K_n edges guarantees a spanning graph G containing all graphs H (v(H)<=n, Delta(H)<=D).
- This guarantee holds when the bias b <= c(n/log n)^(1/D) for D>=2.
- The resulting graph G also contains graphs H with bounded maximum degree and degeneracy <= (D-1)/2.
- The threshold bias is sharp for n/3 disjoint triangles or n/4 disjoint K4 copies.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We prove that for each $$D\ge 2$$ D ≥ 2 there exists $$c>0$$ c > 0 such that whenever $$b\le c\big (\tfrac{n}{\log n}\big )^{1/D}$$ b ≤ c ( n log n ) 1 / D , in the (1 : b ) Maker-Breaker game played on $$E(K_n)$$ E ( K n ) , Maker has a strategy to guarantee claiming a graph G containing copies of all graphs H with $$v(H)\le n$$ v ( H ) ≤ n and $$\Delta (H)\le D$$ Δ ( H ) ≤ D . We show further that the graph G guaranteed by this strategy also contains copies of any graph H with bounded maximum degree and degeneracy at most $$\tfrac{D-1}{2}$$ D - 1 2 . This lower bound on the threshold bias is sharp up to the $$\log $$ log -factor when H consists of $$\tfrac{n}{3}$$ n 3 vertex-disjoint triangles or $$\tfrac{n}{4}$$ n 4 vertex-disjoint $$K_4$$ K 4 -copies.
The authors' abstract, as published at the source. COMBINATORICA, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics