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Concept
Balaban 10-cage
Summary
In the mathematical field of graph theory, the Balaban 10-cage or Balaban (3,10)-cage is a 3-regular graph with 70 vertices and 105 edges named after Alexandru T. Balaban. Published in 1972, It was the first 10-cage discovered but it is not unique.
The complete list of 10-cages and the proof of minimality was given by Mary R. O'Keefe and Pak Ken Wong. There exist 3 distinct (3,10)-cages, the other two being the Harries graph and the Harries–Wong graph. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.
The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-vertex-connected graph and 3-edge-connected. The book thickness is 3 and the queue number is 2.
The characteristic polynomial of the Balaban 10-cage is
Image:balaban_10-cage_2COL.svg|The [[chromatic number]] of the Balaban 10-cage is 2.
Image:balaban_10-cage_3color_edge.svg|The [[chromatic index]] of the Balaban 10-cage is 3.
Image: balaban_10-cage_alternative_drawing.svg|Another drawing of the Balaban 10-cage.
Official source
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