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The harmonious chromatic number of a graph \(G\), denoted \(h(G)\), is the smallest number of colors needed to color the vertices of \(G\) so that adjacent vertices receive different colors and no two edges have the same pair of colors represented at their endvertices.
The mixed harmonious Ramsey number \(H(a, b)\) is defined to be the smallest integer \(p\) such that if a graph \(G\) has \(p\) vertices, then either \(h(G) \geq a\) or \(\alpha(G) \geq b\). For certain values of \(a\) and \(b\), we determine the exact value of \(H(a,b)\). In some other cases, we are able to determine upper and lower bounds for \(H(a, b)\).