Dual Connectivity in Graphs
An edge-coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> of a connected graph <i>G</i> is called rainbow if there exists...
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MDPI AG
2025-01-01
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| Series: | Mathematics |
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| author | Mohammed A. Mutar Daniele Ettore Otera Hasan A. Khawwan |
| author_facet | Mohammed A. Mutar Daniele Ettore Otera Hasan A. Khawwan |
| author_sort | Mohammed A. Mutar |
| collection | DOAJ |
| description | An edge-coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> of a connected graph <i>G</i> is called rainbow if there exists a rainbow path connecting any pair of vertices. In contrast, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is monochromatic if there is a monochromatic path between any two vertices. Some graphs can admit a coloring which is simultaneously rainbow and monochromatic; for instance, any coloring of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> is rainbow and monochromatic. This paper refers to such a coloring as dual coloring. We investigate dual coloring on various graphs and raise some questions about the sufficient conditions for connected graphs to be dual connected. |
| format | Article |
| id | doaj-art-405daff90e4842f3bfe9763cc532041e |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
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| series | Mathematics |
| spelling | doaj-art-405daff90e4842f3bfe9763cc532041e2025-01-24T13:39:49ZengMDPI AGMathematics2227-73902025-01-0113222910.3390/math13020229Dual Connectivity in GraphsMohammed A. Mutar0Daniele Ettore Otera1Hasan A. Khawwan2Department of Mathematics, College of Science, University of Al Qadisiyah, Diwaniyah 58001, IraqInstitute of Data Science and Digital Technologies, Faculty of Mathematics and Informatics, Vilnius University, 08412 Vilnius, LithuaniaEducation Directorate of Al-Qadisiyah, Diwaniyah 58001, IraqAn edge-coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> of a connected graph <i>G</i> is called rainbow if there exists a rainbow path connecting any pair of vertices. In contrast, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is monochromatic if there is a monochromatic path between any two vertices. Some graphs can admit a coloring which is simultaneously rainbow and monochromatic; for instance, any coloring of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> is rainbow and monochromatic. This paper refers to such a coloring as dual coloring. We investigate dual coloring on various graphs and raise some questions about the sufficient conditions for connected graphs to be dual connected.https://www.mdpi.com/2227-7390/13/2/229edge coloringrainbow coloringmonochromatic coloringdual connected graphs |
| spellingShingle | Mohammed A. Mutar Daniele Ettore Otera Hasan A. Khawwan Dual Connectivity in Graphs Mathematics edge coloring rainbow coloring monochromatic coloring dual connected graphs |
| title | Dual Connectivity in Graphs |
| title_full | Dual Connectivity in Graphs |
| title_fullStr | Dual Connectivity in Graphs |
| title_full_unstemmed | Dual Connectivity in Graphs |
| title_short | Dual Connectivity in Graphs |
| title_sort | dual connectivity in graphs |
| topic | edge coloring rainbow coloring monochromatic coloring dual connected graphs |
| url | https://www.mdpi.com/2227-7390/13/2/229 |
| work_keys_str_mv | AT mohammedamutar dualconnectivityingraphs AT danieleettoreotera dualconnectivityingraphs AT hasanakhawwan dualconnectivityingraphs |