| Summary: | A subgraph <i>H</i> of an edge-colored graph <i>G</i> is called rainbow if all of its edges have different colors. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> denote the maximum positive integer <i>t</i>, such that there is a <i>t</i>-edge-colored graph <i>G</i> without any rainbow subgraph <i>H</i>. We denote by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><msub><mi>K</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> a matching of size <i>k</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">O</mi><mi>n</mi></msub></semantics></math></inline-formula> the class of all maximal outer-planar graphs on <i>n</i> vertices, respectively. The outer-planar anti-Ramsey number of graph <i>H</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><mrow><mi>a</mi><mi>r</mi></mrow><mrow><mo>(</mo><msub><mi>O</mi><mi>n</mi></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>|</mo><mspace width="3.33333pt"></mspace><msub><mi>O</mi><mi>n</mi></msub><mo>∈</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>. It seems nontrivial to determine the exact values for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>a</mi><mi>r</mi></mrow><mo>(</mo><msub><mi mathvariant="bold-script">O</mi><mi>n</mi></msub><mo>,</mo><mi>k</mi><msub><mi>K</mi><mn>2</mn></msub><mo>)</mo><mo>≤</mo><mi>n</mi><mo>+</mo><mn>3</mn><mi>k</mi><mo>−</mo><mn>8</mn></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>6</mn></mrow></semantics></math></inline-formula>, which improves the existing upper bound for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>,</mo><mi>k</mi><msub><mi>K</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>, and prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>,</mo><mi>k</mi><msub><mi>K</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>5</mn></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></semantics></math></inline-formula>. We also obtain that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>r</mi><mo>(</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>,</mo><mn>6</mn><msub><mi>K</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mn>6</mn></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>29</mn></mrow></semantics></math></inline-formula>.
|
|---|