Planar Graphs without Cycles of Length 3, 4, and 6 are (3, 3)-Colorable

For non-negative integers <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:msub><a:mrow><a:mi>d</a:mi></a:mrow><a:mrow><a:mn>1</a:mn></a:mrow></a:msub></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:msub><c:mrow><c:mi>d</c:mi></c:mrow><c:mrow><c:mn>2</c:mn></c:mrow></c:msub></c:math> , if <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:msub><e:mrow><e:mi>V</e:mi></e:mrow><e:mrow><e:mn>1</e:mn></e:mrow></e:msub></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"><g:msub><g:mrow><g:mi>V</g:mi></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msub></g:math> are two partitions of a graph <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M5"><i:mi>G</i:mi></i:math> ’s vertex set <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M6"><k:mi>V</k:mi><k:mfenced open="(" close=")" separators="|"><k:mrow><k:mi>G</k:mi></k:mrow></k:mfenced></k:math> , such that <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M7"><p:msub><p:mrow><p:mi>V</p:mi></p:mrow><p:mrow><p:mn>1</p:mn></p:mrow></p:msub></p:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M8"><r:msub><r:mrow><r:mi>V</r:mi></r:mrow><r:mrow><r:mn>2</r:mn></r:mrow></r:msub></r:math> induce two subgraphs of <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M9"><t:mi>G</t:mi></t:math> , called <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M10"><v:mi>G</v:mi><v:mfenced open="[" close="]" separators="|"><v:mrow><v:msub><v:mrow><v:mi>V</v:mi></v:mrow><v:mrow><v:mn>1</v:mn></v:mrow></v:msub></v:mrow></v:mfenced></v:math> with maximum degree at most <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M11"><ab:msub><ab:mrow><ab:mi>d</ab:mi></ab:mrow><ab:mrow><ab:mn>1</ab:mn></ab:mrow></ab:msub></ab:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M12"><cb:mi>G</cb:mi><cb:mfenced open="[" close="]" separators="|"><cb:mrow><cb:msub><cb:mrow><cb:mi>V</cb:mi></cb:mrow><cb:mrow><cb:mn>2</cb:mn></cb:mrow></cb:msub></cb:mrow></cb:mfenced></cb:math> with maximum degree at most <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M13"><hb:msub><hb:mrow><hb:mi>d</hb:mi></hb:mrow><hb:mrow><hb:mn>2</hb:mn></hb:mrow></hb:msub></hb:math> , respectively, then the graph <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" id="M14"><jb:mi>G</jb:mi></jb:math> is said to be improper <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M15"><lb:mfenced open="(" close=")" separators="|"><lb:mrow><lb:msub><lb:mrow><lb:mi>d</lb:mi></lb:mrow><lb:mrow><lb:mn>1</lb:mn></lb:mrow></lb:msub><lb:mo>,</lb:mo><lb:msub><lb:mrow><lb:mi>d</lb:mi></lb:mrow><lb:mrow><lb:mn>2</lb:mn></lb:mrow></lb:msub></lb:mrow></lb:mfenced></lb:math> -colorable, as well as <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M16"><qb:mfenced open="(" close=")" separators="|"><qb:mrow><qb:msub><qb:mrow><qb:mi>d</qb:mi></qb:mrow><qb:mrow><qb:mn>1</qb:mn></qb:mrow></qb:msub><qb:mo>,</qb:mo><qb:msub><qb:mrow><qb:mi>d</qb:mi></qb:mrow><qb:mrow><qb:mn>2</qb:mn></qb:mrow></qb:msub></qb:mrow></qb:mfenced></qb:math> -colorable. A class of planar graphs without <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M17"><vb:msub><vb:mrow><vb:mi>C</vb:mi></vb:mrow><vb:mrow><vb:mn>3</vb:mn></vb:mrow></vb:msub><vb:mo>,</vb:mo><vb:msub><vb:mrow><vb:mi>C</vb:mi></vb:mrow><vb:mrow><vb:mn>4</vb:mn></vb:mrow></vb:msub></vb:math> , and <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" id="M18"><xb:msub><xb:mrow><xb:mi>C</xb:mi></xb:mrow><xb:mrow><xb:mn>6</xb:mn></xb:mrow></xb:msub></xb:math> is denoted by <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" id="M19"><zb:mi mathvariant="script">C</zb:mi></zb:math> . In 2019, Dross and Ochem proved that <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" id="M20"><cc:mi>G</cc:mi></cc:math> is <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" id="M21"><ec:mfenced open="(" close=")" separators="|"><ec:mrow><ec:mn>0</ec:mn><ec:mo>,</ec:mo><ec:mn>6</ec:mn></ec:mrow></ec:mfenced></ec:math> -colorable, for each graph <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" id="M22"><jc:mi>G</jc:mi></jc:math> in <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" id="M23"><lc:mi mathvariant="script">C</lc:mi></lc:math> . Given that <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" id="M24"><oc:msub><oc:mrow><oc:mi>d</oc:mi></oc:mrow><oc:mrow><oc:mn>1</oc:mn></oc:mrow></oc:msub><oc:mo>+</oc:mo><oc:msub><oc:mrow><oc:mi>d</oc:mi></oc:mrow><oc:mrow><oc:mn>2</oc:mn></oc:mrow></oc:msub><oc:mo>≥</oc:mo><oc:mn>6</oc:mn></oc:math> , this inspires us to investigate whether <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" id="M25"><qc:mi>G</qc:mi></qc:math> is <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" id="M26"><sc:mfenced open="(" close=")" separators="|"><sc:mrow><sc:msub><sc:mrow><sc:mi>d</sc:mi></sc:mrow><sc:mrow><sc:mn>1</sc:mn></sc:mrow></sc:msub><sc:mo>,</sc:mo><sc:msub><sc:mrow><sc:mi>d</sc:mi></sc:mrow><sc:mrow><sc:mn>2</sc:mn></sc:mrow></sc:msub></sc:mrow></sc:mfenced></sc:math> -colorable, for each graph <xc:math xmlns:xc="http://www.w3.org/1998/Math/MathML" id="M27"><xc:mi>G</xc:mi></xc:math> in <zc:math xmlns:zc="http://www.w3.org/1998/Math/MathML" id="M28"><zc:mi mathvariant="script">C</zc:mi></zc:math> . In this paper, we provide a partial solution by showing that <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" id="M29"><cd:mi>G</cd:mi></cd:math> is (3, 3)-colorable, for each graph <ed:math xmlns:ed="http://www.w3.org/1998/Math/MathML" id="M30"><ed:mi>G</ed:mi></ed:math> in <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" id="M31"><gd:mi mathvariant="script">C</gd:mi></gd:math> .