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Keywords:
extremal graph problem; similarity of graphs
extremal graph problem; similarity of graphs
Summary:
In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g(n,l)$, the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$. By edit distance of two graphs $G$, $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$. This new extremal number $g(n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show that $g(n, l)$ is close to $\frac 12\binom n2$ for small values of $l>2$. We also present some exact values for small $n$ and lower bounds for very large $l$ close to the number of non-isomorphic graphs of $n$ vertices.
References:
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