Sparse multipartite graphs as partition universal for graphs with bounded degree

Qizhong Lin () and Yusheng Li ()
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Qizhong Lin: Fuzhou University
Yusheng Li: Tongji University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 3, No 4, 724-739

Abstract: Abstract For graphs G and H, let $$G\rightarrow (H,H)$$ G → ( H , H ) signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote $$\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}$$ H ( Δ , n ) = { H : | V ( H ) | = n , Δ ( H ) ≤ Δ } . For any $$\Delta $$ Δ and n, we say that G is partition universal for $$\mathcal {H}(\Delta ,n)$$ H ( Δ , n ) if $$G\rightarrow (H,H)$$ G → ( H , H ) for every $$H\in \mathcal {H}(\Delta ,n)$$ H ∈ H ( Δ , n ) . Let $$G_r(N,p)$$ G r ( N , p ) be the random spanning subgraph of the complete r-partite graph $$K_r(N)$$ K r ( N ) with N vertices in each part, in which each edge of $$K_r(N)$$ K r ( N ) appears with probability p independently and randomly. We prove that for fixed $$\Delta \ge 2$$ Δ ≥ 2 there exist constants r, B and C depending only on $$\Delta $$ Δ such that if $$N\ge Bn$$ N ≥ B n and $$p=C(\log N/N)^{1/\Delta }$$ p = C ( log N / N ) 1 / Δ , then asymptotically almost surely $$G_r(N,p)$$ G r ( N , p ) is partition universal for $$\mathcal {H}(\Delta ,n)$$ H ( Δ , n ) .

Keywords: Partition universal; Sparse regularity lemma; Probabilistic method (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0214-1

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