, uv to T i , part T i remains a good cactus (that all of w 1 , w 2 , w 3 belong to P , which means that at least two of w 1 , w 2 , w 3 are internal vertices of P , one of which is not adjacent to w 4 in P . Assume w 1 is an internal vertex of P not adjacent, P reaches u via w 4 ). If, say, w 1 , does not belong to P , then, by adding both w 1 u

, Again, if w 4 u does not belong to T i , then we are done as earlier

. .. , y 5 the five neighbours of v different from u. If d = 1, i.e., G is connected, then note that G is a planar graph with even size |E(G)| ? 10

, Note that this adds a path of length 2 that is pendant or isolated in T 5 , so it remains an even forest, ? If T 5 does not contain u, then we add uv and vy 1 to T 5

, Since d ? 2, there are y i , y j that belong to different connected components of G . Then we add uv, vy i , vy j to T 1 . Note that T 1 remains a good cactus, since the three edges we have added are not involved in any cycle of T 1 . Furthermore, these edges belong to a connected component with maximum degree at least 3, which is thus not an odd-length path

, Similarly as for other existing proofs, our proofs involve decompositions into auxiliary structures that are to be further decomposed. As a consequence, most known bounds, including ours, are still far from the conjectured one in Conjecture 1.1, even for very particular classes of graphs. Still, we believe that the decomposition methods we have, Conclusion In this work, we have improved known upper bounds on the irregular chromatic index of some families of degenerate graphs

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