Displaying similar documents to “An asymptotic result for the path partition conjecture.”

A path(ological) partition problem

Izak Broere, Michael Dorfling, Jean E. Dunbar, Marietjie Frick (1998)

Discussiones Mathematicae Graph Theory

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Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.

Detour chromatic numbers

Marietjie Frick, Frank Bullock (2001)

Discussiones Mathematicae Graph Theory

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The nth detour chromatic number, χₙ(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χₙ(G) ≤ ⎡(τ(G))/n⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.

A ramsey-type theorem for multiple disjoint copies of induced subgraphs

Tomoki Nakamigawa (2014)

Discussiones Mathematicae Graph Theory

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Let k and ℓ be positive integers with ℓ ≤ k − 2. It is proved that there exists a positive integer c depending on k and ℓ such that every graph of order (2k−1−ℓ/k)n+c contains n vertex disjoint induced subgraphs, where these subgraphs are isomorphic to each other and they are isomorphic to one of four graphs: (1) a clique of order k, (2) an independent set of order k, (3) the join of a clique of order ℓ and an independent set of order k − ℓ, or (4) the union of an independent set of...

Hamiltonicity of k -traceable graphs.

Bullock, Frank, Dankelmann, Peter, Frick, Marietjie, Henning, Michael A., Oellermann, Ortrud R., van Aardt, Susan (2011)

The Electronic Journal of Combinatorics [electronic only]

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Heavy subgraph pairs for traceability of block-chains

Binlong Li, Hajo Broersma, Shenggui Zhang (2014)

Discussiones Mathematicae Graph Theory

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A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two...