Geodesic
On graphs with vertices of two colors and groups with 3-transpositions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 1, pp. 257-262

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We consider labeled undirected graphs without loops or multiple edges with vertices of two colors. A coloring of a graph $\Gamma_n$ is called odd-connected if the removal of vertices of the first color produces an odd number of connected components. A general formula for the number $t_n$ of odd-connected colorings is found for certain families of embedded graphs $\Gamma_n$. The formula depends on two parameters. In the cases where two graphs in a family can be interpreted as Coxeter graphs for certain groups with 3-transpositions, specific formulas for the numbers $t_n$ are found.
Keywords: labeled graph, graph coloring, generating function, Сoxeter graph, group with 3-transpositions. 
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A. I. Sozutov; I. O. Aleksandrova. On graphs with vertices of two colors and groups with 3-transpositions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 1, pp. 257-262. http://geodesic.mathdoc.fr/item/TIMM_2016_22_1_a24/