On Computing the Path Number of a Graph

Botler, Fábio; Cano, Rafael G.; Sambinelli, Maycon

doi:10.1016/j.entcs.2019.08.017

Cited by 1 publication

(2 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such graphs are known as odd semicliques [1]. This motivates the following strengthening of Conjecture 1, which was considered in [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…They also prove [4] that a graph with maximum degree at most 4 is either a Gallai graph, or one of the three odd semicliques with maximum degree at most 4 (

K_{3}

K_{5} - e

, and

K_{5}

). More recently, Botler et al [6] verified Conjecture 4 for triangle‐free planar graphs by proving that every such graph is a Gallai graph. In this paper, we explore an intermediate statement between Conjectures 1 and 4.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Towards Gallai's path decomposition conjecture

Botler

Sambinelli

2020

Journal of Graph Theory

View full text Add to dashboard Cite

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.

show abstract

“…Such graphs are known as odd semicliques [1]. This motivates the following strengthening of Conjecture 1, which was considered in [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…They also prove [4] that a graph with maximum degree at most 4 is either a Gallai graph, or one of the three odd semicliques with maximum degree at most 4 (