Routing Optimality for Demand Graphs of Matching Number at Most Two

We prove that network coding and fractional routing have the same capacity region in undirected networks whose demand graph has matching number at most two, that is, when no three demand edges have pairwise disjoint endpoints. The metric-packing reduction of Liu et al. reduces this problem to a distance-weighted communication inequality on the fixed graph $Γ_{3,3}$. We first prove two entropy inequalities for sessions between designated terminal pairs on $Γ_{3,3}$. We then extend this result to sessions with arbitrary endpoints to obtain the distance-weighted communication bound required by the reduction. The proof combines conditional entropy inequalities with two graph contractions and an information inequality for deterministic interaction. Combining this inequality with the existing reduction establishes routing optimality for this class of demand graphs.