| 2422 | |
| 2423 | def ErreraGraph(self): |
| 2424 | r""" |
| 2425 | Returns the Errera graph. |
| 2426 | |
| 2427 | For more information, see this |
| 2428 | `Wikipedia article on the Errera graph <http://en.wikipedia.org/wiki/Errera_graph>`_. |
| 2429 | |
| 2430 | EXAMPLES: |
| 2431 | |
| 2432 | The Errera graph is named after Alfred Errera. It is a planar graph |
| 2433 | on 17 vertices and having 45 edges. :: |
| 2434 | |
| 2435 | sage: G = graphs.ErreraGraph(); G |
| 2436 | Errera graph: Graph on 17 vertices |
| 2437 | sage: G.is_planar() |
| 2438 | True |
| 2439 | sage: G.order() |
| 2440 | 17 |
| 2441 | sage: G.size() |
| 2442 | 45 |
| 2443 | |
| 2444 | The Errera graph is Hamiltonian with radius 3, diameter 4, girth 3, |
| 2445 | and chromatic number 4. :: |
| 2446 | |
| 2447 | sage: G.is_hamiltonian() |
| 2448 | True |
| 2449 | sage: G.radius() |
| 2450 | 3 |
| 2451 | sage: G.diameter() |
| 2452 | 4 |
| 2453 | sage: G.girth() |
| 2454 | 3 |
| 2455 | sage: G.chromatic_number() |
| 2456 | 4 |
| 2457 | |
| 2458 | Each vertex degree is either 5 or 6. That is, if `f` counts the |
| 2459 | number of vertices of degree 5 and `s` counts the number of vertices |
| 2460 | of degree 6, then `f + s` is equal to the order of the Errera |
| 2461 | graph. :: |
| 2462 | |
| 2463 | sage: D = G.degree_sequence() |
| 2464 | sage: D.count(5) + D.count(6) == G.order() |
| 2465 | True |
| 2466 | |
| 2467 | The automorphism group of the Errera graph is isomorphic to the |
| 2468 | dihedral group of order 20. :: |
| 2469 | |
| 2470 | sage: ag = G.automorphism_group() |
| 2471 | sage: ag.is_isomorphic(DihedralGroup(10)) |
| 2472 | True |
| 2473 | """ |
| 2474 | edge_dict = { |
| 2475 | 0: [1,7,14,15,16], |
| 2476 | 1: [2,9,14,15], |
| 2477 | 2: [3,8,9,10,14], |
| 2478 | 3: [4,9,10,11], |
| 2479 | 4: [5,10,11,12], |
| 2480 | 5: [6,11,12,13], |
| 2481 | 6: [7,8,12,13,16], |
| 2482 | 7: [13,15,16], |
| 2483 | 8: [10,12,14,16], |
| 2484 | 9: [11,13,15], |
| 2485 | 10: [12], |
| 2486 | 11: [13], |
| 2487 | 13: [15], |
| 2488 | 14: [16]} |
| 2489 | return graph.Graph(edge_dict, name="Errera graph") |