The size of uniquely colorable graphs

@article{Shaoji1990TheSO,
  title={The size of uniquely colorable graphs},
  author={Xu Shaoji},
  journal={Journal of Combinatorial Theory, Series B},
  year={1990},
  volume={50},
  pages={319-320}
}
  • Xu Shaoji
  • Published 1 December 1990
  • Mathematics
  • Journal of Combinatorial Theory, Series B
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19 Citations
Highly Influential Citations
1
Background Citations
4
Methods Citations
1

19 Citations

Algebraic characterization of uniquely vertex colorable graphs
A construction of uniquely colourable graphs with equal colour class sizes
Uniquely colorable Cayley graphs
TLDR
It is proved that ( k  − 1) n is a sharp lower bound for the number of edges of a uniquely k -colorable, noncomplete Cayley graph over an abelian group of order n.
Constructions of Uniquely 3-Colorable Graphs
  • Zepeng Li, Jin Xu
  • Mathematics
    2016 IEEE First International Conference on Data Science in Cyberspace (DSC)
  • 2016
TLDR
This paper gives a simple counterexample on 16 vertices to S J Xu's conjecture concerning uniquely 3-colorable graphs without triangles, and based on this graph they construct an infinite family of uniquely3- colorable, 4-regular and triangle-free graphs by their construction methods.
Coloring Graphs Having Few Colorings Over Path Decompositions
TLDR
A Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw( G)poly(n)s time, that distinguishes between k- Colorable graphs having at most s proper k-colorings and non-k-colorable graphs.
Uniquely Coloring Graphs over Path Decompositions
TLDR
A new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form is exploited and shows that if the authors instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex, many orientations give a non-zero value if the graph has a unique $k$-coloring.
The Relationships between Wiener Index, Stability Number and Clique Number of Composite Graphs
Some new relations have been established between Wiener indices, stability numbers and clique numbers for several classes of composite graphs that arise via graph products. For three of considered…
On two generalizations of the Alon-Tarsi polynomial method
GRAPH CHOOSABILITY AND DOUBLE LIST COLORABILITY
In this paper, we give a sufficient condition for graph choosability, based on Combinatorial Nullstellensatz and a specific property, called "double list colorability", which means that there is a…
Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph
The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first…
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References

Uniquely colorable graphs