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Fractional Gallai–Edmonds decomposition and maximal graphs on fractional matching number
Journal of Combinatorial Optimization ( IF 1 ) Pub Date : 2020-04-09 , DOI: 10.1007/s10878-020-00566-4
Yan Liu , Mengxia Lei , Xueli Su

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, \(\sum \nolimits _{e\in \Gamma (v)}f(e) \le 1\), where \(\Gamma (v)\) is the set of all edges incident with v. The fractional matching number \(\mu _{f}(G)\) of G is the supremum of \(\sum \nolimits _{e\in E(G)}f(e)\) over all fractional matchings f of G. Let \(D_f(G)\) be the set of vertices which are unsaturated by some maximum fractional matching of G, \(A_f(G)\) the set of vertices in \(V(G)-D_f(G)\) adjacent to a vertex in \(D_f(G)\) and \(C_f(G)=V(G)-A_f(G)-D_f(G)\). In this paper, the partition \((C_f(G), A_f(G), D_f(G))\), named fractional Gallai–Edmonds decomposition, is obtained by an algorithm in polynomial time via the Gallai–Edmonds decomposition. A graph G is maximal on \(\mu _{f}(G)\) if any addition of edge increases the fractional matching number \(\mu _{f}(G)\). 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.

中文翻译:

分数Gallai–Edmonds分解和分数匹配数的最大图

G的分数匹配是一个函数f,它为[0,1]中的每个边分配一个数字,使得对于每个顶点v\(\ sum \ nolimits _ {e \ in \ Gamma(v)} f( e)\ le 1 \),其中\(\ Gamma(v)\)是与v入射的所有边的集合。分数匹配数\(\亩_ {F}(G)\)G ^是上确界\(\总和\ nolimits _ {ê\在E(G)} F(E)\)在所有分数匹配数˚Fģ。令\(D_f(G)\)为通过G的最大分数匹配不饱和的一组顶点,\(A_f(G)\)在该组顶点的\(V(G)-D_f(G)\)邻近于一个顶点\(D_F(G)\)\(C_F(G)= V(G)-A_f(G)-D_f (G)\)。在本文中,分区\((C_f(G),A_f(G),D_f(G))\),被称为分数Gallai–Edmonds分解,是通过多项式时间内的算法通过Gallai–Edmonds分解获得的。的图G ^是对最大\(\亩_ {F}(G)\)如果任何另外的边缘增加了分数匹配数\(\亩_ {F}(G)\)。图兰数是最大图的边数的最大值,饱和数是最大图的边数的最小值。在本文中,最大图通过使用分数Gallai–Edmonds分解来表征。这样就得到了图兰数,饱和度数和极值图。
更新日期:2020-04-09
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