| Summary: | A theta graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Θ</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula> is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> of <i>G</i> is a proper total coloring of <i>G</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><msub><mi>E</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>u</mi><mo stretchy="false">}</mo></mrow></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>≠</mo><msub><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><msub><mi>E</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo></mrow></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for each edge <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> denotes the set of edges incident with a vertex <i>u</i>. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> admits an NSD total <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="sans-serif">Δ</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mo>≥</mo><mn>9</mn></mrow></semantics></math></inline-formula> but without theta graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Θ</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula> by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.
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