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In this paper, we present, as we are aware of, the firstcombinatorialalgorithm specifically designed for the minimum weightedintegercoloring problem ({\it MWIP}). We test thealgorithm on randomly generated graphs with integer weights uniformlydrawn from intervals [1, 1], [1, 2], [1,5], [1, 10], [1, 15], and [1, 20].We also use theproposed algorithm to test the quality of a simple,yet effectiveheuristic for the {\it MWIPS} in theliterature. We have observed from our test that: i( thealgorithm is able to solve {\it MWIPS} on graphs of upto 20 vertices when the average vertex weights is not too large;ii( The relative gap between the simple heuristic solutions andthe optimal solution seems to decrease as the average vertex weightincreases. A rough comparison with the state-of-the-art methods forthe minimum unweighted coloring problem seems to suggest theadvantage of solving {\it MWIPS} directly.