edge colourings
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Author(s):  
Olivia Reade

AbstractWe introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of $$T_{k,l}$$ T k , l , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.



2020 ◽  
Vol 8 ◽  
Author(s):  
Stefan Ehard ◽  
Stefan Glock ◽  
Felix Joos

Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.



10.37236/6303 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Katherine Edwards ◽  
António Girão ◽  
Jan Van den Heuvel ◽  
Ross J. Kang ◽  
Gregory J. Puleo ◽  
...  

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.



COMBINATORICA ◽  
2018 ◽  
Vol 38 (5) ◽  
pp. 1265-1267
Author(s):  
Tamás Fleiner
Keyword(s):  


2018 ◽  
Vol 69 ◽  
pp. 185-199
Author(s):  
Jakub Przybyło
Keyword(s):  


Author(s):  
Urszula Bednarz ◽  
Iwona Włoch
Keyword(s):  

In this paper we determine successive extremal trees with respect to the number of all \((A,2B)\)-edge colourings.



2017 ◽  
Vol 127 ◽  
pp. 205-214 ◽  
Author(s):  
Florian Lehner
Keyword(s):  


2017 ◽  
Vol 61 ◽  
pp. 199-205 ◽  
Author(s):  
Pilar Cano ◽  
Guillem Perarnau ◽  
Oriol Serra


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