The Handshaking Lemma states that in any finite graph, the sum of the degrees of all vertices equals twice the number of edges, and consequently that the number of vertices with odd degree is always even. It is one of the most elementary results of graph theory, so named because it also describes the impossibility of an odd number of people at a gathering each shaking hands with an odd number of others.
Facts
StatementIn every finite undirected graph, the number of vertices with odd degree is even, because the sum of all vertex degrees equals twice the number of edges. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Handshaking Lemma (Wikipedia)
Sources
1. Handshaking Lemma (Wikipedia)
Wikimedia FoundationLead paragraph, opening definition
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even.
Lead paragraph, attribution sentence
Both results were proven by Leonhard Euler (1736) in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory.
Lead section, statement-form reference
For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even.
In Branch: Graph Theory, Lead sentence
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch
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