In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, the Meredith graph is a 4-

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

undirected graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...

with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973. The Meredith graph is 4- vertex-connected and 4- edge-connected, has

chromatic number In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

chromatic index In graph theory, a proper edge coloring of a Graph (discrete mathematics), graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge colorin ...

5, radius 7, diameter 8, girth 4 and is non-Hamiltonian. It has

book thickness In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a ''book'', a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on ...

3 and

queue number In the mathematical field of graph theory, the queue number of a Graph (discrete mathematics), graph is a graph invariant defined analogously to book thickness, stack number (book thickness) using Queue (abstract data type), first-in first-out (q ...

2. Published in 1973, it provides a counterexample to the

Crispin Nash-Williams Crispin St John Alvah Nash-Williams FRSE (19 December 1932 – 20 January 2001) was a British mathematician. His research interest was in the field of discrete mathematics, especially graph theory. Biography Nash-Williams was born on 19 Decemb ...

conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian. However,

W. T. Tutte William Thomas Tutte (; 14 May 1917 – 2 May 2002) was an English and Canadian code breaker and mathematician. During the Second World War, he made a fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system ...

showed that all 4-connected

planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...

s are hamiltonian.Tutte, W.T., ed., Recent Progress in Combinatorics. Academic Press, New York, 1969. The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

of the Meredith graph is

(x-4) (x-1)^ x^ (x+1)^ (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4

Gallery

Image:Meredith graph 3COL.svg, The

of the Meredith graph is 3. Image:Meredith graph 5color edge.svg, The

of the Meredith graph is 5.

References

{{reflist Individual graphs Regular graphs