Mass–luminosity relation: Difference between revisions
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{{infobox graph | |||
| name = Meredith graph | |||
| image = [[Image:Meredith graph.svg|220px]] | |||
| image_caption = The Meredith graph | |||
| namesake = G. H. Meredith | |||
| vertices = 70 | |||
| edges = 140 | |||
| automorphisms = | |||
| girth = 4 | |||
| diameter = 8 | |||
| radius = 7 | |||
| chromatic_number = 3 | |||
| chromatic_index = 5 | |||
| properties = [[Eulerian graph|Eulerian]] | |||
}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Meredith graph''' is a 4-[[regular graph|regular]] [[undirected graph]] with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.<ref>{{MathWorld|urlname=MeredithGraph|title=Meredith graph}}</ref> | |||
The Meredith graph is 4-[[k-vertex-connected graph|vertex-connected]] and 4-[[k-edge-connected graph|edge-connected]], has [[chromatic number]] 3, [[chromatic index]] 5, radius 7, diameter 8, girth 4 and is [[Hamiltonian graph|non-hamiltonian]].<ref>Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p. 470, 2007.</ref> | |||
Published in 1973, it provides a counterexample to the [[Crispin Nash-Williams]] conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.<ref>Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.</ref><ref>Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.</ref> However, [[W. T. Tutte]] showed that all 4-connected [[planar graph]]s are hamiltonian.<ref>Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.</ref> | |||
The [[characteristic polynomial]] of the Meredith graph is <math>(x-4) (x-1)^{10} x^{21} (x+1)^{11} (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4</math>. | |||
==Gallery== | |||
<gallery> | |||
Image:Meredith graph 3COL.svg|The [[chromatic number]] of the Meredith graph is 3. | |||
Image:Meredith graph 5color edge.svg|The [[chromatic index]] of the Meredith graph is 5. | |||
</gallery> | |||
== References == | |||
{{reflist}} | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] |
Revision as of 01:13, 14 December 2013
In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.[1]
The Meredith graph is 4-vertex-connected and 4-edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-hamiltonian.[2]
Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.[3][4] However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.[5]
The characteristic polynomial of the Meredith graph is .
Gallery
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The chromatic number of the Meredith graph is 3.
-
The chromatic index of the Meredith graph is 5.
References
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- ↑ Meredith, G. H. J. "Regular 4-Valent 4-Connected Nonhamiltonian Non-4-Edge-Colorable Graphs." J. Combin. Th. B 14, 55-60, 1973.
- ↑ Bondy, J. A. and Murty, U. S. R. "Graph Theory with Applications". New York: North Holland, p. 239, 1976.
- ↑ Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969.