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- W1972024351 abstract "If a graph<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>admits a<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-list assignment<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>such that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>has a unique<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>-coloring, then<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>is called uniquely<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-list colorable graph, or U<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>LC graph for short. In the process of characterizing U<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>LC graphs, the complete multipartite graphs<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi mathvariant=normal>*</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>are often researched. But it is usually not easy to construct the unique<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M12><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-list assignment of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M13><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi mathvariant=normal>*</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>. In this paper, we give some propositions about the property of the graph<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M14><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi mathvariant=normal>*</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>when it is U<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M15><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>LC, which provide a very significant guide for constructing such list assignment. Then a special example of U<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M16><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>LC graphs<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M17><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi mathvariant=normal>*</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>as a application of these propositions is introduced. The conclusion will pave the way to characterize U<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M18><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>LC complete multipartite graphs." @default.
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- W1972024351 title "Some Conclusion on Uniquek-List Colorable Complete Multipartite Graphs" @default.
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