Uniquely colorable graph

Uniquely colorable graph

In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) "k"-coloring up to permutation of the colors.

"Example 1". A "minimal imperfect graph" is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph.

Some properties of a uniquely "k"-colorable graph "G" with "n" vertices and "m" edges:
# "m" ≥ ("k" - 1) "n" - "k"("k"-1)/2. (Truszczyński 1981; Xu 1990)

A uniquely edge-colorable graph is a "k"-edge-chromatic graph that has only one possible (proper) "k"-edge-coloring up to permutation of the colors.

"Example 2". The stars "K"1,"k" are uniquely "k"-edge-colorable graphs. Moreover, R. J. Wilson (1967) conjectured and A. G. Thomason (1978) proved that, when "k" ≥ 4, they are also the only members in this family.See [Bollobás 1978] .

A uniquely total colorable graph is a "k"-total-chromatic graph that has only one possible (proper) "k"-total-coloring up to permutation of the colors.

"Example 3". Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs.Mahmoodian and 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"2. (Akbari et al. 1997)
# Δ("G") ≤ 2 δ("G"). (Akbari et al. 1997)
# Δ("G") ≤ n/2 + 1. (Akbari 2003)Here χ″("G") is the total chromatic number; Δ("G"), maximum degree; and δ("G"), minimum degree.

References

* Akbari, S. (2003). Two conjectures on uniquely totally colorable graphs. "Discrete Math." 266(1-3), 41–45.
* Akbari, S.; Behzad, M.; Haijiabolhassen, H.; Mahmoodian (1997). Uniquely total colorable graphs. "Graphs Combin." 13, 305–314.
* Bollobás, Béla (1978). "Extremal graph theory", Vol. 11, LMS Monographs. London; New York; San Francisco: Academic Press.
* Hillar, C.; Windfeldt, T., An algebraic characterization of uniquely vertex colorable graphs, Journal of Combinatorial Theory Series B, 98 (2008), 400–414.
* Mahmoodian, E. S.; Shokrollahi, M. A. (1995). "Open problems at the combinatorics workshop of AIMC25 (Tehran, 1994)", in Combinatorics Advances. In Colbourn, C. J.; Mahmoodian, E. S. (Eds.), Mathematics and its applications, 321–324. Dordrecht; Boston; London: Kluwer Academic Publishers.
* Truszczyński, M. (1981). "Some results on uniquely colourable graphs". Soloquia Math. Soc. János Bolyai, 37, 733–746.
* Xu, Shaoji (1990). The size of uniquely colorable graphs. "J. Combin. Theory (B)" 50, 319–320.


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