
Gallai's path decomposition conjecture for trianglefree planar graphs
A path decomposition of a graph G is a collection of edgedisjoint paths...
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Maximal degrees in subgraphs of Kneser graphs
In this paper, we study the maximum degree in nonempty induced subgraph...
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On the Standard (2,2)Conjecture
The wellknown 123 Conjecture asserts that the edges of every graph wi...
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MultiStage Graph Peeling Algorithm for Probabilistic Core Decomposition
Mining dense subgraphs where vertices connect closely with each other is...
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Decomposition of (2k+1)regular graphs containing special spanning 2kregular Cayley graphs into paths of length 2k+1
A P_ℓdecomposition of a graph G is a set of paths with ℓ edges in G tha...
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On sensitivity in bipartite Cayley graphs
Huang proved that every set of more than half the vertices of the ddime...
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Atomic subgraphs and the statistical mechanics of networks
We develop random graph models where graphs are generated by connecting ...
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Towards Gallai's path decomposition conjecture
A path decomposition of a graph G is a collection of edgedisjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2. Seminal results towards its verification consider the graph obtained from G by removing its vertices of odd degree, which is called the Esubgraph of G. Lovász (1968) verified Gallai's Conjecture for graphs whose Esubgraphs consist of at most one vertex, and Pyber (1996) verified it for graphs whose Esubgraphs are forests. In 2005, Fan verified Gallai's Conjecture for graphs in which each block of their Esubgraph is trianglefree and has maximum degree at most 3. Let calG be the family of graphs for which (i) each block has maximum degree at most 3; and (ii) 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 Esubgraphs are subgraphs of graphs in calG. This allows the components of the Esubgraphs to contain any number of blocks with triangles as long as they are subgraphs of graphs in calG.
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