
Subcubic planar graphs of girth 7 are class I
We prove that planar graphs of maximum degree 3 and of girth at least 7 ...
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Elimination distance to bounded degree on planar graphs
We study the graph parameter elimination distance to bounded degree, whi...
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Decomposing 4connected planar triangulations into two trees and one path
Refining a classical proof of Whitney, we show that any 4connected plan...
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Guarding Quadrangulations and Stacked Triangulations with Edges
Let G = (V,E) be a plane graph. A face f of G is guarded by an edge vw ∈...
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Rumors in a Network: Who's the Culprit?
We provide a systematic study of the problem of finding the source of a ...
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Induced and Weak Induced Arboricities
We define the induced arboricity of a graph G, denoted by ia(G), as the...
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Computing kModal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of ...
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Treelike distance colouring for planar graphs of sufficient girth
Given a multigraph G and a positive integer t, the distancet chromatic index of G is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than t edges must receive different colours. Let π'_t(d) and τ'_t(d) be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree d. We have that π'_t(d) is at most and at least a nontrivial constant multiple larger than τ'_t(d). (We conjecture _d→∞π'_2(d)/τ'_2(d) =9/4 in particular.) We prove for odd t the existence of a quantity g depending only on t such that the distancet chromatic index of any planar multigraph of maximum degree d and girth at least g is at most τ'_t(d) if d is sufficiently large. Such a quantity does not exist for even t. We also show a related, similar phenomenon for distance vertexcolouring.
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