Series-parallel graphs and hypercube degeneracy

Abstract

Several recent works have identified patterns that must exist in dense subsets of either the vertices or the edges of a large hypercube. We introduce a framework, based on the concept of series-parallel graphs, that unifies and generalizes these results.

Figures (3)

• Figure 1: \ref{['fig:v']} Multigraph used in the proof of \ref{['cor:alon']}. Labels represent edge multiplicities. \ref{['fig:e']} Multigraph used in the proof of \ref{['cor:partite']}. Labels represent edge multiplicities and the distinguished edge highlighted. \ref{['fig:theta']} A $1$-sum of two copies of $\Theta_4$ that does not have a partite representation.
• Figure 2: Graphs used in the proof of \ref{['prop:fib']}.
• Figure 3: Multigraphs used in the proofs of \ref{['fig:smallv']}\ref{['prop:smallv']}, and \ref{['fig:smalle']}\ref{['prop:smalle']}. The distinguished edge in \ref{['fig:smalle']} is highlighted, and labels give the corresponding indices of a set of parallel edges. The location of the $I_0$ edges is irrelevant to the argument.

Theorems & Definitions (18)

• proof
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• proof : Proof of \ref{['thm:fullv']}
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