On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to the permutation of the colors. For a plane graph G, two faces &...

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Main Authors:	Zepeng Li, Naoki Matsumoto, Enqiang Zhu, Jin Xu, Tommy Jensen
Format:	Article
Language:	English
Published:	MDPI AG 2019-09-01
Series:	Mathematics
Subjects:	plane graph unique coloring uniquely three-colorable plane graph construction adjacent (<i>i</i>,<i>j</i>)-faces
Online Access:	https://www.mdpi.com/2227-7390/7/9/793

id	doaj-cd30f97d87db41148ed119f76710f699
record_format	Article
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Zepeng Li Naoki Matsumoto Enqiang Zhu Jin Xu Tommy Jensen
spellingShingle	Zepeng Li Naoki Matsumoto Enqiang Zhu Jin Xu Tommy Jensen On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees Mathematics plane graph unique coloring uniquely three-colorable plane graph construction adjacent (<i>i</i>,<i>j</i>)-faces
author_facet	Zepeng Li Naoki Matsumoto Enqiang Zhu Jin Xu Tommy Jensen
author_sort	Zepeng Li
title	On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees
title_short	On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees
title_full	On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees
title_fullStr	On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees
title_full_unstemmed	On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees
title_sort	on uniquely 3-colorable plane graphs without adjacent faces of prescribed degrees
publisher	MDPI AG
series	Mathematics
issn	2227-7390
publishDate	2019-09-01
description	A graph <i>G</i> is <i>uniquely k-colorable</i> if the chromatic number of <i>G</i> is <i>k</i> and <i>G</i> has only one <i>k</i>-coloring up to the permutation of the colors. For a plane graph <i>G</i>, two faces <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> of <i>G</i> are <i>adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces</i> if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi></mrow></semantics></math></inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>j</mi></mrow></semantics></math></inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> have a common edge, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is the degree of a face <i>f</i>. In this paper, we prove that every uniquely three-colorable plane graph has adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≤</mo> <mn>5</mn></mrow></semantics></math></inline-formula>. The bound of five for <i>k</i> is the best possible. Furthermore, we prove that there exists a class of uniquely three-colorable plane graphs having neither adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces nor adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </semantics> </math> </inline-formula> are fixed in <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">{</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">}</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi></mrow></semantics></math></inline-formula>. One of our constructions implies that there exists an infinite family of edge-critical uniquely three-colorable plane graphs with <i>n</i> vertices and <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> <mi>n</mi> <mo>-</mo> <mfrac> <mn>14</mn> <mn>3</mn> </mfrac> </mrow> </semantics> </math> </inline-formula> edges, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mo>≥</mo> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is odd and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>≡</mo> <mn>2</mn> <mspace width="4.44443pt"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mn>3</mn> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>.
topic	plane graph unique coloring uniquely three-colorable plane graph construction adjacent (<i>i</i>,<i>j</i>)-faces
url	https://www.mdpi.com/2227-7390/7/9/793
work_keys_str_mv	AT zepengli onuniquely3colorableplanegraphswithoutadjacentfacesofprescribeddegrees AT naokimatsumoto onuniquely3colorableplanegraphswithoutadjacentfacesofprescribeddegrees AT enqiangzhu onuniquely3colorableplanegraphswithoutadjacentfacesofprescribeddegrees AT jinxu onuniquely3colorableplanegraphswithoutadjacentfacesofprescribeddegrees AT tommyjensen onuniquely3colorableplanegraphswithoutadjacentfacesofprescribeddegrees
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spelling	doaj-cd30f97d87db41148ed119f76710f6992020-11-25T02:44:23ZengMDPI AGMathematics2227-73902019-09-017979310.3390/math7090793math7090793On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed DegreesZepeng Li0Naoki Matsumoto1Enqiang Zhu2Jin Xu3Tommy Jensen4School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, ChinaResearch Institute for Digital Media and Content, Keio University, Tokyo 108-8345, JapanInstitute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, ChinaSchool of Electronics Engineering and Computer Science, Peking University, Beijing 100871, ChinaDepartment of Mathematics, Kyungpook National University, Daegu 41566, KoreaA graph <i>G</i> is <i>uniquely k-colorable</i> if the chromatic number of <i>G</i> is <i>k</i> and <i>G</i> has only one <i>k</i>-coloring up to the permutation of the colors. For a plane graph <i>G</i>, two faces <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> of <i>G</i> are <i>adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces</i> if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi></mrow></semantics></math></inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>j</mi></mrow></semantics></math></inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> have a common edge, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is the degree of a face <i>f</i>. In this paper, we prove that every uniquely three-colorable plane graph has adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≤</mo> <mn>5</mn></mrow></semantics></math></inline-formula>. The bound of five for <i>k</i> is the best possible. Furthermore, we prove that there exists a class of uniquely three-colorable plane graphs having neither adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces nor adjacent <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-faces, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </semantics> </math> </inline-formula> are fixed in <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">{</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">}</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi></mrow></semantics></math></inline-formula>. One of our constructions implies that there exists an infinite family of edge-critical uniquely three-colorable plane graphs with <i>n</i> vertices and <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> <mi>n</mi> <mo>-</mo> <mfrac> <mn>14</mn> <mn>3</mn> </mfrac> </mrow> </semantics> </math> </inline-formula> edges, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mo>≥</mo> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is odd and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>≡</mo> <mn>2</mn> <mspace width="4.44443pt"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mn>3</mn> <mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/7/9/793plane graphunique coloringuniquely three-colorable plane graphconstructionadjacent (<i>i</i>,<i>j</i>)-faces

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

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