Perfect bicoloring of the quintic graphs of order at most 10

Résumé

‎In this paper‎, ‎we investigate the problem of finding perfect bicolorings for graphs with degree five and at most 10 vertices‎. ‎A perfect bicoloring is a partition of the vertex set into two subsets such that each subset induces a regular subgraph‎. ‎We use some algebraic techniques to construct parameter matrices that encode the properties of perfect bicolorings‎. ‎We then classify all the possible parameter matrices for graphs with degree five and at most 10 vertices‎, ‎and determine which of them correspond to graphs that admit perfect bicolorings‎.