The coloring of disk graphs is motivated by the frequency assignment problem. In 1998, Malesińska et al. introduced double disk graphs as their generalization. They showed that the chromatic number of a double disk graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation> is at most <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$33\,\omega (G) - 35$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>33</mn> <mspace width="0.166667em"/> <mi mathvariant="italic">ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>35</mn> </mrow> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\omega (G)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> denotes the size of a maximum clique in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation>. Du et al. improved the upper bound to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$31\,\omega (G) - 1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>31</mn> <mspace width="0.166667em"/> <mi mathvariant="italic">ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>. In this paper we decrease the bound substantially; namely we show that the chromatic number of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation> is at most <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$15\,\omega (G) - 14$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>15</mn> <mspace width="0.166667em"/> <mi mathvariant="italic">ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>14</mn> </mrow> </math> </EquationSource> </InlineEquation>. Copyright Springer Science+Business Media New York 2014
