The Harries graph is a (3,10)-cage (a cubic graph of girth 10 with the smallest number of vertices).
The square graph of the line graph of the Harries graph is vertex 7 colorable, and it is not vertex 5 colorable. Is it vertex 6 colorable? I have not been able to determine this with the tools I have at hand - would somebody know?
The edge list of the Harries graph is the following (the numbers are the vertices):
{{1, 2}, {1, 42}, {1, 70}, {2, 3}, {2, 53}, {3,4}, {3, 60}, {4, 5}, {4, 17}, {5, 6}, {5, 26}, {6, 7}, {6, 49}, {7, 8}, {7, 34}, {8, 9}, {8, 41}, {9, 10}, {9, 66}, {10, 11}, {10, 23}, {11, 12}, {11, 30}, {12, 13}, {12, 61}, {13, 14}, {13, 50}, {14, 15}, {14, 43}, {15, 16}, {15, 56}, {16, 17}, {16, 67}, {17, 18}, {18, 19}, {18, 31}, {19, 20}, {19, 40}, {20, 21}, {20, 63}, {21, 22}, {21, 48}, {22, 23}, {22, 55}, {23, 24}, {24, 25}, {24, 37}, {25, 26}, {25, 44}, {26, 27}, {27, 28}, {27, 64}, {28, 29}, {28, 57}, {29, 30}, {29, 70}, {30, 31}, {31, 32}, {32, 33}, {32, 45}, {33, 34}, {33, 54}, {34, 35}, {35, 36}, {35, 62}, {36, 37}, {36, 69}, {37, 38}, {38, 39}, {38, 51}, {39, 40}, {39, 58}, {40, 41}, {41, 42}, {42, 43}, {43, 44}, {44, 45}, {45, 46}, {46, 47}, {46, 59}, {47, 48}, {47, 68}, {48, 49}, {49, 50}, {50, 51}, {51, 52}, {52, 53}, {52, 65}, {53, 54}, {54, 55}, {55, 56}, {56, 57}, {57, 58}, {58, 59}, {59, 60}, {60, 61}, {61, 62}, {62, 63}, {63, 64}, {64, 65}, {65, 66}, {66, 67}, {67, 68}, {68, 69}, {69, 70}}.
I will be grateful if somebody can help with this.
After several days of computation on a 2 year old laptop this was resolved. The chromatic number of the square graph of the line graph of the Harries graph is 7, and not 6.