Subgraph

A subgraph is obtained from a graph by selecting some of its vertices and edges while retaining every selected edge's endpoints.

A spanning subgraph retains every vertex of the original graph and some or all of its edges. Adding the omitted edges can only enlarge the graph neighbourhood and the external vertex boundary of each fixed vertex set.

The subgraph induced by a vertex set $W$ contains every edge of the original graph whose two endpoints lie in $W$.

Every finite graph $G$ has a three-colourable subgraph with at least $2e(G)/3$ edges. Give vertices three independent uniform colours and retain edges whose endpoints have different colours. Each edge survives with probability $2/3$, so the expected number surviving is $2e(G)/3$.