A Generalization of a Theorem of Diderrich in Additive Group Theory to Vertex-transitive Graphs

LetX1, X2, . . . , Xkbe a family of vertex-transitive graphs without triangles on the same vertex-set. We consider the set Λ of vertices which can be reached from a given vertexaby a pathe1 · · · et, whereeiis an edge ofXrj(i)with1≤j(1)<j(2) <· · ·<j(t)≤kand 0≤t≤k(fort=0, Λis equal to the singleton {a}). either Λ contains one of the connected components ofain one of theXi’s, or the cardinality of Λ is at least 1 plus the sum of thekoutdegrees. esult, applied to groups, is a refinement of a theorem of Diderrich in additive group theory [2].