Question

What is the chromatic number of K3,K4 and K5?

Answer 1

What is the chromatic number of K3,K4 and K5?mplete bipartite graph with m = 5 and n = 3

Vertices

n + m

Edges

mn

Radius

{\displaystyle \left\{{\begin{array}{ll}1&m=1\vee n=1\\2&{\text{otherwise}}\end{array}}\right.} \left\{{\begin{array}{ll}1&m=1\vee n=1\\2&{\text{otherwise}}\end{array}}\right.

Diameter

{\displaystyle \left\{{\begin{array}{ll}1&m=n=1\\2&{\text{otherwise}}\end{array}}\right.} \left\{{\begin{array}{ll}1&m=n=1\\2&{\text{otherwise}}\end{array}}\right.

Girth

{\displaystyle \left\{{\begin{array}{ll}\infty &m=1\vee n=1\\4&{\text{otherwise}}\end{array}}\right.} \left\{{\begin{array}{ll}\infty &m=1\vee n=1\\4&{\text{otherwise}}\end{array}}\right.

Automorphisms

{\displaystyle \left\{{\begin{array}{ll}2m!n!&n=m\\m!n!&{\text{otherwise}}\end{array}}\right.} \left\{{\begin{array}{ll}2m!n!&n=m\\m!n!&{\text{otherwise}}\end{array}}\right.

Chromatic number

2

Chromatic index

max{m, n}

Spectrum

{\displaystyle \{0^{n+m-2},(\pm {\sqrt {nm}})^{1}\}} \{0^{n+m-2},(\pm {\sqrt {nm}})^{1}\}

Notation

{\displaystyle K_{m,n}} K_{m,n}