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Let (P;≼) be a partially ordered set (poset, briefly) with a least element 0. In this thesis, we deal with zero-divisor graphs of posets. We show that if the chromatic number r(P) and the clique number r(P) (x(r(P)) and !(r(P)), respectively) are finite, then x(r(P)) = !(w(P)) = n in which n is the number of minimal prime ideals of P. We also prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or 1  

We study colorings of quasiordered sets (qosets) with a least element 0. To any qoset Q with 0 we assign a graph (called a zerodivisor graph) whose vertices are labelled by the elements of Q with two vertices x; y adjacent if the only elements lying below x and y are those lying below 0. We prove that for such graphs, the chromatic number and the clique number coincide  

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x, y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide.Also, we give a condition under which posets are not finitely colorable