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Fractional Gallai–Edmonds decomposition and maximal graphs on fractional matching number

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Fractional Gallai–Edmonds decomposition and maximal graphs on fractional matching number

10$

Yan Liu, Mengxia Lei & Xueli Su 

Abstract

A fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that for each vertex v, ∑e∈Γ(v)f(e)≤1∑e∈Γ(v)f(e)≤1, where Γ(v)Γ(v) is the set of all edges incident with v. The fractional matching number μf(G)μf(G) of G is the supremum of ∑e∈E(G)f(e)∑e∈E(G)f(e) over all fractional matchings f of G. Let Df(G)Df(G) be the set of vertices which are unsaturated by some maximum fractional matching of G, Af(G)Af(G) the set of vertices in V(G)−Df(G)V(G)−Df(G) adjacent to a vertex in Df(G)Df(G) and Cf(G)=V(G)−Af(G)−Df(G)Cf(G)=V(G)−Af(G)−Df(G). In this paper, the partition (Cf(G),Af(G),Df(G))(Cf(G),Af(G),Df(G)), named fractional Gallai–Edmonds decomposition, is obtained by an algorithm in polynomial time via the Gallai–Edmonds decomposition. A graph G is maximal on μf(G)μf(G) if any addition of edge increases the fractional matching number μf(G)μf(G). The Turán number is the maximum of edge numbers of maximal graphs and the saturation number is the minimum of edge numbers of maximal graphs. In this paper, the maximal graphs are characterized by using the fractional Gallai–Edmonds decomposition. Thus the Turán number, saturation number and extremal graphs are obtained.

Only units of this product remain
Year 2020
Language English
Format PDF
DOI 10.1007/s10878-020-00566-4