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$K_5$ and $K_{3,3}$ illustrate complete and complete bipartite graphs, degrees and edge counts

Example

The complete graph $K_5$ has degree $4$ at every vertex and has $10$ edges. The complete bipartite graph $K_{3,3}$ has degree $3$ at every vertex and has $9$ edges.

Facts & Assumptions

Given: The standard graphs $K_5$ and $K_{3,3}$ .

[F1]

$K_5$ contains every pair of its five vertices as an edge, while $K_{3,3}$ has two three-element parts and every edge between the parts (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[L1]

$K_5$ has $\binom52$ edges (The complete graph on an $n$ -element vertex set has $\binom{n}{2}$ edges).

[L2]

The sum of degrees is twice the number of edges (Handshake lemma: the sum of the vertex degrees is twice the number of edges).

Verification

technique · direct

1.1

Each vertex of $K_5$ is adjacent to the other four vertices, so its degree is $4$ ; [L1] gives $|E(K_5)|=\binom52=10$ .

F1L1algebra

1.2

Each vertex of $K_{3,3}$ is adjacent to all three vertices in the opposite part and none in its own part, so all six vertices have degree $3$ .

2.1

By [L2], $2|E(K_{3,3})|=6\cdot3=18$ , hence $|E(K_{3,3})|=9$ .

step 1.2L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)

K5K_5K5​ and K3,3K_{3,3}K3,3​ illustrate complete and complete bipartite graphs, degrees and edge counts