v n ) containing all other vertices of G. Since H is spanning, connected, and unicyclic, |E(H)| = |V (G)|, which is at most |E(G)| ? 3, since |E(G)| ? |V (G)| + 3. All conditions are now met to invoke the arguments in the proof of Theorem 3.5, from which we can deduce a proper {0, 1, 2}-labelling of G where adjacent vertices get distinct colours modulo 3 and in which only the edges of (our) H are possibly assigned label 0. Let us now consider the subgraph H of G obtained from H by adding the edge v n v 1 , which is present in G. Recall that (v n v 1 ) = 2 by default. Note that H contains at least two disjoint perfect matchings M 1 , M 2 . Indeed, since |V (G)| is even, a first perfect matching M 1 of H contains v 1 v 2, Let us consider H, the subgraph of G containing the x edges of C x , and all the (other) edges of the Hamiltonian cycle ,
, + 1, which is less than |E(G)|/3 since |E(G)| ? |V (G)| + 3
, However, it covers all such graphs with at least three chords. Thus, to get a generalisation of Theorem 4.18 for all 2-connected outerplanar graphs, it remains to prove a similar result for the 3-chromatic ones with at most two chords. Those with no chords are exactly odd-length cycles, for which the claim holds (see
This allows us to use our tools from Section 3 to establish an ? 6, we have that mT(G) = 0. Indeed, let v be the center of the star T , and let v 2 , . . . , v n be the leaves of T . We can construct a proper 2-labelling of G as follows: start from v 2 v 3 and, following the edges of the cycle joining the leaves of T in increasing order of their indices ,
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