| Summary: | The strong chromatic index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>χ</mo><mi mathvariant="normal">s</mi><mo>′</mo></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a graph <i>G</i> is the minimum number of colors needed in a proper edge-coloring so that every color class induces a matching in <i>G</i>. It was proved In 2013, that every outerplanar graph <i>G</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>χ</mo><mi mathvariant="normal">s</mi><mo>′</mo></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn><mo>Δ</mo><mo>−</mo><mn>3</mn></mrow></semantics></math></inline-formula>. In this paper, we give a characterization for an outerplanar graph <i>G</i> to have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>χ</mo><mi mathvariant="normal">s</mi><mo>′</mo></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mo>Δ</mo><mo>−</mo><mn>3</mn></mrow></semantics></math></inline-formula>. We also show that if <i>G</i> is a bipartite outerplanar graph, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>χ</mo><mi mathvariant="normal">s</mi><mo>′</mo></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mo>Δ</mo></mrow></semantics></math></inline-formula>; and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>χ</mo><mi mathvariant="normal">s</mi><mo>′</mo></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mo>Δ</mo></mrow></semantics></math></inline-formula> if and only if <i>G</i> contains a particular subgraph.
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