Editing Uniquely colorable graph (section)

===Unique total colorability===
A '''uniquely total colorable graph''' is a [[Total coloring|''k''-total-chromatic graph]] that has only one possible [[Total coloring|(proper) ''k''-total-coloring]] up to permutation of the colors.

[[Empty graph]]s, [[Path graph|paths]], and [[Cycle graph|cycles]] of length divisible by 3 are uniquely total colorable graphs.
{{harvtxt|Mahmoodian|Shokrollahi|1995}} conjectured that they are also the only members in this family.

Some properties of a uniquely ''k''-total-colorable graph ''G'' with ''n'' vertices:
# χ″(''G'') = Δ(''G'') + 1 unless ''G'' = ''K''<sub>2</sub>.{{sfnp|Akbari|Behzad|Hajiabolhassan|Mahmoodian|1997}}
# Δ(''G'') ≤ 2 δ(''G'').{{sfnp|Akbari|Behzad|Hajiabolhassan|Mahmoodian|1997}}
# Δ(''G'') ≤ n/2 + 1.{{sfnp|Akbari|2003}}
Here χ″(''G'') is the [[Total coloring|total chromatic number]]; Δ(''G'') is the [[Glossary of graph theory#Degree|maximum degree]]; and δ(''G'') is the [[Glossary of graph theory#Degree|minimum degree]].