Gallai's Path Decomposition for 2-degenerate Graphs
Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at...
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Discrete Mathematics & Theoretical Computer Science
2023-05-01
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| Online Access: | http://dmtcs.episciences.org/10313/pdf |
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| author | Nevil Anto Manu Basavaraju |
| author_facet | Nevil Anto Manu Basavaraju |
| author_sort | Nevil Anto |
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| description | Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle. |
| format | Article |
| id | doaj-art-1dffe0bb6c0c40eba240b21f12435983 |
| institution | Kabale University |
| issn | 1365-8050 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj-art-1dffe0bb6c0c40eba240b21f124359832025-08-20T03:42:37ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502023-05-01vol. 25:1Graph Theory10.46298/dmtcs.1031310313Gallai's Path Decomposition for 2-degenerate GraphsNevil Antohttps://orcid.org/0000-0002-3379-1377Manu BasavarajuGallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.http://dmtcs.episciences.org/10313/pdfmathematics - combinatoricscomputer science - discrete mathematics05c38, 05c70g.2.2 |
| spellingShingle | Nevil Anto Manu Basavaraju Gallai's Path Decomposition for 2-degenerate Graphs Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics computer science - discrete mathematics 05c38, 05c70 g.2.2 |
| title | Gallai's Path Decomposition for 2-degenerate Graphs |
| title_full | Gallai's Path Decomposition for 2-degenerate Graphs |
| title_fullStr | Gallai's Path Decomposition for 2-degenerate Graphs |
| title_full_unstemmed | Gallai's Path Decomposition for 2-degenerate Graphs |
| title_short | Gallai's Path Decomposition for 2-degenerate Graphs |
| title_sort | gallai s path decomposition for 2 degenerate graphs |
| topic | mathematics - combinatorics computer science - discrete mathematics 05c38, 05c70 g.2.2 |
| url | http://dmtcs.episciences.org/10313/pdf |
| work_keys_str_mv | AT nevilanto gallaispathdecompositionfor2degenerategraphs AT manubasavaraju gallaispathdecompositionfor2degenerategraphs |