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- W2787260064 abstract "A graph $G$ is said to have textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| leq b$ whenever ${i,j}$ is an edge of $G$. Recently, Bottcher, Schacht, and Taraz verified a conjecture of Bollobas and Komlos which says that for every positive $r,Delta,gamma$, there exists $beta$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $Delta$ which has bandwidth at most $beta n$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + gamma)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend this theorem to dense random graphs. For bipartite $H$, this answers an open question of Bottcher, Kohayakawa, and Taraz. It appears that for non-bipartite $H$ the direct extension is not possible, and one needs in addition that some vertices of $H$ have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r + gamma)np$." @default.
- W2787260064 created "2018-02-23" @default.
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- W2787260064 date "2010-05-11" @default.
- W2787260064 modified "2023-09-27" @default.
- W2787260064 title "Bandwidth theorem for sparse graphs" @default.
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