Seymour’s Second Neighborhood Conjecture for m-Free Oriented Graphs

<jats:p>Let <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>D</mi> <mo>=</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>V</mi> <mo>,</mo> <mi>E</mi> </mrow> </mfenced> </mrow> </mfenced> </math> </jats:inline-formula> be an oriented graph with minimum out-degree <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M2"> <msup> <mrow> <mi>δ</mi> </mrow> <mrow> <mo>+</mo> </mrow> </msup> </math> </jats:inline-formula>. For <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M3"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>D</mi> </mrow> </mfenced> </math> </jats:inline-formula>, let <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M4"> <msubsup> <mi>d</mi> <mi>D</mi> <mo>+</mo> </msubsup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M5"> <msubsup> <mi>d</mi> <mi>D</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> </jats:inline-formula> be the out-degree and second out-degree of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M6"> <mi>x</mi> </math> </jats:inline-formula> in <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M7"> <mi>D</mi> </math> </jats:inline-formula>, respectively. For a directed graph <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M8"> <mi>D</mi> </math> </jats:inline-formula>, we say that a vertex <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M9"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>D</mi> </mrow> </mfenced> </math> </jats:inline-formula> is a Seymour vertex if <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M10"> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi>d</mi> <mi>D</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>≥</mo> <msubsup> <mi>d</mi> <mi>D</mi> <mo>+</mo> </msubsup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </mrow> </mfenced> </math> </jats:inline-formula>. Seymour in 1990 conjectured that each oriented graph has a Seymour vertex. A directed graph <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M11"> <mi>D</mi> </math> </jats:inline-formula> is called <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M12"> <mi>m</mi> </math> </jats:inline-formula>-free if there are no directed cycles with length at most <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M13"> <mi>m</mi> </math> </jats:inline-formula> in <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M14"> <mi>D</mi> </math> </jats:inline-formula>. A directed graph <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M15"> <mi>D</mi> <mo>=</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>V</mi> <mo>,</mo> <mi>E</mi> </mrow> </mfenced> </math> </jats:inline-formula> is called <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M16"> <mi>k</mi> </math> </jats:inline-formula>-transitive if, for any directed <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M17"> <mi>x</mi> <mi>y</mi> </math> </jats:inline-formula>-path of length <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M18"> <mi>k</mi> </math> </jats:inline-formula>, there exists <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M19"> <mfenced open="(" close=")" separators="|"> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </mfenced> <mo>∈</mo> <mi>E</mi> </mrow> </mfenced> </math> </jats:inline-formula>. In this paper, we show that (1) each <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M20"> <mfenced open="(" close=")" separators="|"> <mrow> <msup> <mrow> <mi>δ</mi> </mrow> <mrow> <mo>+</mo> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> </math> </jats:inline-formula>-free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M21"> <mi>m</mi> </math> </jats:inline-formula>-free and <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M22"> <mfenced open="(" close=")" separators="|">