King Graph
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A002943, A140521, and A158651 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
1-3-2022
2360
King Graph
The 
king graph is a graph with 
vertices in which each vertex represents a square in an 
chessboard, and each edge corresponds to a legal move by a king.
The number of edges in the 
king graph is 
, so for 
, 2, ..., the first few values are 0, 6, 20, 42, 72, 110, ... (OEIS A002943).
The order 
graph has chromatic number 
for 
and 
for 
. For 
, 3, ..., the edge chromatic numbers are 3, 8, 8, 8, 8, ....
King graphs are implemented in the Wolfram Language as GraphData[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/KingGraph/Inline13.svg" style="height:22px; width:6px" />"King", ![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/KingGraph/Inline14.svg" style="height:22px; width:6px" />m, n![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/KingGraph/Inline15.svg" style="height:22px; width:6px" />![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/KingGraph/Inline16.svg" style="height:22px; width:6px" />].
All king graphs are Hamiltonian and biconnected. The only regular king graph is the 
-king graph, which is isomorphic to the tetrahedral graph 
. The 
-king graphs are planar only for 
(with the 
case corresponding to path graphs) and 
, some embeddings of which are illustrated above.
The 
-king graph is perfect iff 
(S. Wagon, pers. comm., Feb. 22, 2013).
Closed formulas for the numbers 
of 
-cycles of 
with 
are given by
where the formula for 
appears in Perepechko and Voropaev.
The numbers of Hamiltonian cycles for the 
-king graphs for 
, 3, ... are 6, 32, 5660, 4924128, ... (OEIS A140521), with the corresponding numbers of Hamiltonian paths given by 24, 784, 343184, ... (OEIS A158651).
REFERENCES
Karavaev, A. M. "FlowProblem: Statistics of Simple Cycles." http://flowproblem.ru/paths/statistics-of-simple-cycles.
Perepechko, S. N. and Voropaev, A. N. "The Number of Fixed Length Cycles in an Undirected Graph. Explicit Formulae in Case of Small Lengths."
Sloane, N. J. A. Sequences A002943, A140521, and A158651 in "The On-Line Encyclopedia of Integer Sequences."
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