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Let $k\\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\\leq i \\leq r$ every vertex in $V_i$ has at most $\\max\\{k, |V_i|-k \\}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\\leq i\\leq r$. 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King","submitted_at":"2010-09-30T22:41:37Z","abstract_excerpt":"Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\\leq i \\leq r$ every vertex in $V_i$ has at most $\\max\\{k, |V_i|-k \\}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\\leq i\\leq r$. 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