Edge-Chromatic Numbers of Glued Graphs

Let G(1) and G(2) be any two graphs. Assume that H-1 subset of G(1) and H-2 subset of G(2) are connected, not a single vertex and such that H-1 congruent to H-2 with an isomorphism f. The glued graph of G(1) and G(2) at H-1 and H-2 with respect to f, denoted by G(1) (sic)(H1 congruent to f H2) G(2) is the graph that results from combining G(1) with G(2) by identifying H-1 and H-2 with respect to the isomorphism f between H-1 and H-2. We give upper bounds of the edge-chromatic numbers of glued graphs; one is in terms of the edge-chromatic numbers of their original graphs where we give a characterization of graphs satisfying its equality. We further obtain a better upper bound of the chromatic numbers of glued graphs when the original graphs are line graphs.