With minimum degree n/2+Θ(m), graphs whose Hamilton cycles all have color-bias below m must be close to a color-dominant or a symmetric 3-partite construction, and this degree condition is tight for r=2.
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Optimal stability results on color-biased Hamilton cycles
With minimum degree n/2+Θ(m), graphs whose Hamilton cycles all have color-bias below m must be close to a color-dominant or a symmetric 3-partite construction, and this degree condition is tight for r=2.