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Let  G  be a bridgeless cubic graph. Consider a list of  k  1‐factors of  G . Let    E  i    be the set of edges contained in precisely  i  members of the  k  1‐factors. Letμ  k    (  G  )be the smallest|    E  0    |over all lists of  k  1‐factors of  G . We study lists by three 1‐factors, and call    G  [   E  0   ∪   E  2   ∪   E  3   ]    with|    E  0    |  =    μ  3    (  G  )aμ  3    (  G  )    ‐core of  G . If  G  is not 3‐edge‐colorable, thenμ  3    (  G  )   ≥  3   . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that ifμ  3    (  G  )   ≠  0   , then    2   μ  3    (  G  )is an upper bound for the girth of  G . We show thatμ  3    (  G  )bounds the oddness    ω  (  G  )    of  G  as well. We prove that    ω   (  G  )   ≤   2  3    μ  3    (  G  )    . If    ω   (  G  )   =   2  3    μ  3    (  G  )    , then everyμ  3    (  G  )    ‐core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4‐edge‐connected cubic graph  G  with    ω   (  G  )   =   2  3    μ  3    (  G  )    . On the other hand, the difference between    ω  (  G  )    and2  3    μ  3    (  G  )can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer    k  ≥  3   , there exists a bridgeless cubic graph  G  such thatμ  3    (  G  )   =  k   .

Petersen Cores and the Oddness of Cubic Graphs