title: Balaban 10-cage
text: # Balaban 10-cage
  
  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
  
      $(x-3)(x-2)(x-1)^{8}x^{2}(x+1)^{8}(x+2)(x+3)\cdot$
      $\cdot (x^{2}-6)^{2}(x^{2}-5)^{4}(x^{2}-2)^{2}(x^{4}-6x^{2}+3)^{8}.$
  
  ## Gallery
  
  * The chromatic number of the Balaban 10-cage is 2.
  * The chromatic index of the Balaban 10-cage is 3.
  * Another drawing of the Balaban 10-cage.
  