Abstract

 
 
 Using the definition of colouring of $2$-edge-coloured graphs derived from 2-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to 2-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those graphs that admit a 2-edge-colouring for which the chromatic polynomial of the graph and the chromatic polynomial of the 2-edge-colouring is equal. Finally, we examine the behaviour of the roots of this polynomial, highlighting behaviours not seen in chromatic polynomials of graphs.
 
 
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