Measurable versions of Vizing's theorem

Abstract

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Approximate Schreier decorations and approximate Kőnig's line coloring Theorem

“…Once we have this then we take the minimal κ 0 < ℵ 1 such that η(A κ 0 ) = 0 and define d = d κ 0 . Note that such κ 0 < ℵ 1 exists by ( 4) and (5). By the Claim we have η(d −1 (a)) ≤ 1 L < ǫ.…”

“…It follows from [5,Theorem 1.8] that χ ′ App (G) ≤ ∆(G) + 1 for any bounded degree Borel graph G. This is the corresponding approximate version of Vizing's Theorem. Since we use this result in the proof of the approximate version of Kőnig line coloring Theorem (Theorem 1.2 (I)) we would like to stress that its proof is significantly easier than the main result of [5,Theorem 1.6].…”

“…It follows from [5,Theorem 1.8] that χ ′ App (G) ≤ ∆(G) + 1 for any bounded degree Borel graph G. This is the corresponding approximate version of Vizing's Theorem. Since we use this result in the proof of the approximate version of Kőnig line coloring Theorem (Theorem 1.2 (I)) we would like to stress that its proof is significantly easier than the main result of [5,Theorem 1.6]. In particular, the combinatorial idea reflects the proof of the Vizing's Theorem for finite graphs in the same way as the proof of Theorem 1.2 (I) reflects the proof of Kőnig's line coloring Theorem for finite bipartite graphs.…”

“…Before we formulate corollaries of the preceding result we would like to point out that the proof of the approximate version of Vizing's Theorem, see [5, Theorem 1.8, Section 5], follows the same strategy as the proof of Proposition 3.2. Namely, modify a given coloring such that it does not contain short weighted Vizing chains, see [5,Section 2.4]. Similarly as in the preceding paragraph, there is an improvement that works simultaneously for every E-invariant measure.…”

“…Let µ be a Borel probability measure on (V, B). By the approximate measurable version of Vizing's Theorem, see [5,Theorem 1.8], we find a proper partial edge coloring c; E → (∆(G) + 1) such that…”