Edge-colorability of graph bundles

The topological notion of a fibre bundle is a generalization both of a Cartesian product and of a covering space. A graph bundle is a combinatorial analog of a fibre bundle. Accordingly, it is a generalization both of a Cartesian product of two graphs and of a covering graph. A “total graph”X is formed from a “base graph”B and “fibre”F. The edge-colorability ofX is studied in terms ofB andF. In particular, it is proved that if a graph bundle with baseB and fibreF satisfies at least one of the conditions:(i) B is of chromatic class 1 andΔ(B) > 0, or (ii) F is of chromatic class 1 andΔ(F) > 0, or (iii) B andF both contain a 1-factor, then its total graphX is of chromatic class 1.