By Rudenskaya O. G.
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Additional resources for 3-Quasiperiodic functions on graphs and hypergraphs
Suppose there exist C ∈ C and c ∈ C such that C := C − c has no 1-factor. By the induction hypothesis (and the fact that, as shown earlier, for ﬁxed G our theorem implies Tutte’s theorem) there exists a set S ⊆ V (C ) with q(C − S ) > |S | . Since |C| is odd and hence |C | is even, the numbers q(C − S ) and |S | are either both even or both odd, so they cannot diﬀer by exactly 1. We may therefore sharpen the above inequality to q(C − S ) giving dC (S ) |S | + 2 , 2. 2 Matching in general graphs where the ﬁrst ‘−1’ comes from the loss of C as an odd component and the second comes from including c in the set T .
1, G contains an Euler tour v0 e0 . . e −1 v , with v = v0 . We replace every vertex v by a pair (v − , v + ), and every − edge ei = vi vi+1 by the edge vi+ vi+1 (Fig. 4). 3 it has a 1-factor. Collapsing every vertex pair (v − , v + ) back into a single vertex v, we turn this 1factor of G into a 2-factor of G. 1) v− v v+ Fig. 4. 2 Matching in general graphs Given a graph G, let us denote by CG the set of its components, and by q(G) the number of its odd components, those of odd order. If G has a 1-factor, then clearly q(G − S) |S| for all S ⊆ V (G), since every odd component of G − S will send a factor edge to S.
Then call a matching M in G stable if for every edge e ∈ E M there exists an edge f ∈ M such that e and f have a common vertex v with e
3-Quasiperiodic functions on graphs and hypergraphs by Rudenskaya O. G.