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Acyclic colourings of planar graphs with large girth
 J. London Math. Soc
, 1999
"... A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ ..."
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A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ
Planar digraphs without large acyclic sets
"... Abstract Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most . When ..."
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Abstract Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most . When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
Maximum 4degenerate subgraph of a planar graph
"... A graph G is kdegenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree k or less. We prove that every connected planar graph with average degree d> 2 has a 4degenerate induced subgraph containing at least (38 − d)/36 of its vertices. This shows that ..."
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A graph G is kdegenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree k or less. We prove that every connected planar graph with average degree d> 2 has a 4degenerate induced subgraph containing at least (38 − d)/36 of its vertices. This shows that every planar graph of order n has a 4degenerate induced subgraph of order more than 8/9 · n. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.
Treecolorable maximal planar graphs 1
"... A treecoloring of a maximal planar graph is a proper vertex 4coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph G is treecolorable if G has a treecoloring. In this article, we prove that a treecolorable maximal planar graph G with δ(G) ≥ ..."
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A treecoloring of a maximal planar graph is a proper vertex 4coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph G is treecolorable if G has a treecoloring. In this article, we prove that a treecolorable maximal planar graph G with δ(G) ≥ 4 contains at least four oddvertices. Moreover, for a treecolorable maximal planar graph of minimum degree 4 that contains exactly four oddvertices, we show that the subgraph induced by its four oddvertices is not a claw and contains no triangles.