Gallai’s path decomposition conjecture for triangle-free planar graphs

Botler, Fábio; Jiménez, Andrea; Sambinelli, Maycon

doi:10.1016/j.disc.2019.01.020

Cited by 12 publications

(7 citation statements)

References 6 publications

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“…Note that these graphs are proper subclasses of 3-degenerate graphs. Hence proving the conjecture for 3-degenerate graphs would generalize the results in Botler et al (2020) and Botler et al (2019).…”

Section: Discussionsupporting

confidence: 59%

“…We have proved Gallai's path decomposition conjecture for 2-degenerate graphs. Botler et al (2020) proved the conjecture for graphs with treewidth at most 3 and Botler et al (2019) proved the conjecture for triangle-free planar graphs. Note that these graphs are proper subclasses of 3-degenerate graphs.…”

Section: Discussionmentioning

confidence: 97%

“…Bonamy and Perrett (2019) asked if the edges of every connected graph G on n vertices can be decomposed into at most ⌊ n 2 ⌋ paths unless G is an odd semi-clique. Botler et al (2020) proved that the edges of any connected graph G with treewidth at most 3 can be decomposed into at most ⌊ n 2 ⌋ paths unless G is K 3 or K 5 − e. Botler et al (2019) proved that the edges of triangle-free planar 2…”

Section: Introductionmentioning

confidence: 99%

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Gallai's Path Decomposition for 2-degenerate Graphs

Nevil¹,

Basavaraju²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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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.

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Section: Discussionsupporting

confidence: 59%

Section: Discussionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Gallai's Path Decomposition for 2-degenerate Graphs

Nevil¹,

Basavaraju²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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“…Moreover, we believe that the technique presented here to deal with this conjecture can be extended for other classes of graphs, for example, graphs with treewidth at most

4

. Recently, by extending the definition of reducing subgraph, Botler, Jiménez, and Sambinelli verified Conjecture for triangle‐free planar graphs.…”

Section: Discussionmentioning

confidence: 93%

Gallai's path decomposition conjecture for graphs with treewidth at most 3

Botler

Sambinelli

Coelho

et al. 2019

Journal of Graph Theory

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A path decomposition of a graph G is a set of edge‐disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph with n vertices admits a path decomposition of size at most ⌊(n+1)∕2⌋. Gallai's conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified Gallai's conjecture for graphs with maximum degree at most 5. In this paper, we verify Gallai's conjecture for graphs with treewidth at most 3. Moreover, we show that the only graphs with treewidth at most 3 that do not admit a path decomposition of size at most ⌊n∕2⌋ are isomorphic to K3 or K5−, the graph obtained from K5 by removing an edge.

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“…Conjecture 1 has been deeply explored, and the literature indicating its correctness includes results for Eulerian graphs with maximum degree at most 4 [10]; a family of regular graphs [5]; a family of triangle‐free graphs [12]; and maximal outerplanar graphs and 2‐connected outerplanar graphs [11]. Recent results were obtained by Bonamy and Perrett [1] who verified Conjecture 1 for graphs with maximum degree at most 5.…”

Section: Introductionmentioning

confidence: 99%

Towards Gallai's path decomposition conjecture

Botler

Sambinelli

2020

Journal of Graph Theory

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A path decomposition of a graph G is a collection of edge‐disjoint paths of G that covers the edge set of G. Gallai conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌈ n / 2 ⌉. Seminal results toward its verification consider the graph obtained from G by removing its vertices of odd degree, which is called the E‐subgraph of G. Lovász verified Gallai's Conjecture for graphs whose E‐subgraphs consist of at most one vertex, and Pyber verified it for graphs whose E‐subgraphs are forests. In 2005, Fan verified Gallai's Conjecture for graphs in which each block, that is, each maximal 2‐connected subgraph, of their E‐subgraph is triangle‐free and has maximum degree at most 3. Let MJX-tex-caligraphicscriptG be the family of graphs for which (a) each block has maximum degree at most 3; and (b) each component either has maximum degree at most 3 or has at most one block that contains triangles. In this paper, we generalize Fan's result by verifying Gallai's Conjecture for graphs whose E‐subgraphs are subgraphs of graphs in MJX-tex-caligraphicscriptG. This allows the components of the E‐subgraphs to contain any number of blocks with triangles as long as they are subgraphs of graphs in MJX-tex-caligraphicscriptG.

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Gallai’s path decomposition conjecture for triangle-free planar graphs

Cited by 12 publications

References 6 publications

Gallai's Path Decomposition for 2-degenerate Graphs

Gallai's Path Decomposition for 2-degenerate Graphs

Gallai's path decomposition conjecture for graphs with treewidth at most 3

Towards Gallai's path decomposition conjecture

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