List colourings of planar graphs
Discrete Mathematics
You can't paint yourself into a corner
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B
Extending graph colorings using no extra colors
Discrete Mathematics
A note on planar 5-list colouring: non-extendability at distance 4
Discrete Mathematics
Precoloring Extensions of Brooks' Theorem
SIAM Journal on Discrete Mathematics
Extending precolorings to circular colorings
Journal of Combinatorial Theory Series B
Distance constraints in graph color extensions
Journal of Combinatorial Theory Series B
Precoloring Extension for 2-connected Graphs
SIAM Journal on Discrete Mathematics
Extending colorings of locally planar graphs
Journal of Graph Theory
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In this paper, we consider coloring of graphs under the assumption that some vertices are already colored. Let G be an r-colorable graph and let P@?V(G). Albertson (1998) has proved that if every pair of vertices in P has distance at least four, then every (r+1)-coloring of G[P] can be extended to an (r+1)-coloring of G, where G[P] is the subgraph of G induced by P. In this paper, we allow P to have pairs of vertices of distance at most three, and investigate how the number of such pairs affects the number of colors we need to extend the coloring of G[P]. We also study the effect of pairs of vertices of distance at most two, and extend the result by Albertson and Moore (1999).