1687 | | The default embedding is an attempt to emphasize the graph's 8 (!!!) |
1688 | | different orbits. In order to understand this better, one can |
1689 | | picture the graph as built in the following way : |
1690 | | |
1691 | | #. One first create a 3-dimensional cube (8 vertices, 12 edges), |
1692 | | whose vertices define the first orbit of the final graph. |
1693 | | |
1694 | | #. The edges of this graph are subdivided once, to create 12 new |
1695 | | vertices which define a second orbit. |
1696 | | |
1697 | | #. The edges of the graph are subdivided once more, to create 24 |
1698 | | new vertices giving a third orbit. |
1699 | | |
1700 | | #. 4 vertices are created and made adjacent to the vertices of |
1701 | | the second orbit so that they have degree 3. These 4 vertices |
1702 | | also define a new orbit. |
1703 | | |
1704 | | #. In order to make the vertices from the third orbit 3-regular |
1705 | | (they all miss one edge), one creates a binary tree on 1 + 3 |
1706 | | + 6 + 12 vertices. The leaves of this new tree are made |
1707 | | adjacent to the 12 vertices of the third orbit, and the graph |
1708 | | is now 3-regular. This binary tree contributes 4 new orbits |
1709 | | to the Harries-Wong graph. |
| 1687 | The default embedding is an attempt to emphasize the graph's |
| 1688 | 8 (!!!) different orbits. In order to understand this better, |
| 1689 | one can picture the graph as being built in the following way: |
| 1690 | |
| 1691 | #. One first creates a 3-dimensional cube (8 vertices, 12 |
| 1692 | edges), whose vertices define the first orbit of the |
| 1693 | final graph. |
| 1694 | |
| 1695 | #. The edges of this graph are subdivided once, to create 12 |
| 1696 | new vertices which define a second orbit. |
| 1697 | |
| 1698 | #. The edges of the graph are subdivided once more, to |
| 1699 | create 24 new vertices giving a third orbit. |
| 1700 | |
| 1701 | #. 4 vertices are created and made adjacent to the vertices |
| 1702 | of the second orbit so that they have degree |
| 1703 | 3. These 4 vertices also define a new orbit. |
| 1704 | |
| 1705 | #. In order to make the vertices from the third orbit |
| 1706 | 3-regular (they all miss one edge), one creates a binary |
| 1707 | tree on 1 + 3 + 6 + 12 vertices. The leaves of this new |
| 1708 | tree are made adjacent to the 12 vertices of the third |
| 1709 | orbit, and the graph is now 3-regular. This binary tree |
| 1710 | contributes 4 new orbits to the Harries-Wong graph. |
1755 | | d[66] = (-9.5,0) |
1756 | | _line_embedding(g, [37, 65,67], first=(-8,2.25), last=(-8,-2.25)) |
1757 | | _line_embedding(g, [36, 38, 64, 24, 68, 30], first=(-7,3), last=(-7,-3)) |
1758 | | _line_embedding(g, [35, 39, 63, 25, 59, 29, 11, 5, 55, 23, 69, 31], first=(-6,3.5), last = (-6,-3.5)) |
1759 | | |
1760 | | # Cube, corners : [9, 15, 21, 27, 45, 51, 57, 61] |
1761 | | _circle_embedding(g, [61, 9], center = (0,-1.5), shift = .2, radius = 4) |
1762 | | _circle_embedding(g, [27, 15], center = (0,-1.5), shift = .7, radius = 4*.707) |
1763 | | _circle_embedding(g, [51, 21], center = (0,2.5), shift = .2, radius = 4) |
1764 | | _circle_embedding(g, [45, 57], center = (0,2.5), shift = .7, radius = 4*.707) |
| 1756 | d[66] = (-9.5, 0) |
| 1757 | _line_embedding(g, [37, 65, 67], first=(-8, 2.25), |
| 1758 | last=(-8, -2.25)) |
| 1759 | _line_embedding(g, [36, 38, 64, 24, 68, 30], first=(-7, 3), |
| 1760 | last=(-7, -3)) |
| 1761 | _line_embedding(g, [35, 39, 63, 25, 59, 29, 11, 5, 55, 23, 69, 31], |
| 1762 | first=(-6, 3.5), last=(-6, -3.5)) |
| 1763 | |
| 1764 | # Cube, corners: [9, 15, 21, 27, 45, 51, 57, 61] |
| 1765 | _circle_embedding(g, [61, 9], center=(0, -1.5), shift=.2, |
| 1766 | radius=4) |
| 1767 | _circle_embedding(g, [27, 15], center=(0, -1.5), shift=.7, |
| 1768 | radius=4*.707) |
| 1769 | _circle_embedding(g, [51, 21], center=(0, 2.5), shift=.2, |
| 1770 | radius=4) |
| 1771 | _circle_embedding(g, [45, 57], center=(0, 2.5), shift=.7, |
| 1772 | radius=4*.707) |
1767 | | _line_embedding(g, [21, 22, 43, 44, 45], first=d[21], last = d[45]) |
1768 | | _line_embedding(g, [21, 4, 3, 56, 57], first=d[21], last = d[57]) |
1769 | | _line_embedding(g, [57, 12, 13, 14, 15], first=d[57], last = d[15]) |
1770 | | _line_embedding(g, [15, 6, 7, 8, 9], first=d[15], last = d[9]) |
1771 | | _line_embedding(g, [9, 10, 19, 20, 21], first=d[9], last = d[21]) |
1772 | | _line_embedding(g, [45, 54, 53, 52, 51], first=d[45], last = d[51]) |
1773 | | _line_embedding(g, [51, 50, 49, 58, 57], first=d[51], last = d[57]) |
1774 | | _line_embedding(g, [51, 32, 33, 34, 61], first=d[51], last = d[61]) |
1775 | | _line_embedding(g, [61, 62, 41, 40, 27], first=d[61], last = d[27]) |
1776 | | _line_embedding(g, [9, 0, 1, 26, 27], first=d[9], last = d[27]) |
1777 | | _line_embedding(g, [27, 28, 47, 46, 45], first=d[27], last = d[45]) |
1778 | | _line_embedding(g, [15, 16, 17, 60, 61], first=d[15], last = d[61]) |
| 1775 | _line_embedding(g, [21, 22, 43, 44, 45], first=d[21], last=d[45]) |
| 1776 | _line_embedding(g, [21, 4, 3, 56, 57], first=d[21], last=d[57]) |
| 1777 | _line_embedding(g, [57, 12, 13, 14, 15], first=d[57], last=d[15]) |
| 1778 | _line_embedding(g, [15, 6, 7, 8, 9], first=d[15], last=d[9]) |
| 1779 | _line_embedding(g, [9, 10, 19, 20, 21], first=d[9], last=d[21]) |
| 1780 | _line_embedding(g, [45, 54, 53, 52, 51], first=d[45], last=d[51]) |
| 1781 | _line_embedding(g, [51, 50, 49, 58, 57], first=d[51], last=d[57]) |
| 1782 | _line_embedding(g, [51, 32, 33, 34, 61], first=d[51], last=d[61]) |
| 1783 | _line_embedding(g, [61, 62, 41, 40, 27], first=d[61], last=d[27]) |
| 1784 | _line_embedding(g, [9, 0, 1, 26, 27], first=d[9], last=d[27]) |
| 1785 | _line_embedding(g, [27, 28, 47, 46, 45], first=d[27], last=d[45]) |
| 1786 | _line_embedding(g, [15, 16, 17, 60, 61], first=d[15], last=d[61]) |