Precoloring extension. I: Interval graphs
Discrete Mathematics - Special volume (part 1) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs”
List colourings of planar graphs
Discrete Mathematics
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Color-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
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
Extending precolorings of subgraphs of locally planar graphs
European Journal of Combinatorics - Special issue: Topological graph theory
Precoloring Extensions of Brooks' Theorem
SIAM Journal on Discrete Mathematics
Extending precolorings to circular colorings
Journal of Combinatorial Theory Series B
Extending colorings of locally planar graphs
Journal of Graph Theory
Precoloring extension involving pairs of vertices of small distance
Discrete Applied Mathematics
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Let G be an r-chromatic graph with an s-colorable subgraph, each of whose components is s-colored with (possibly different) s colors taken from a set of r+s colors. Then if the components of the precolored subgraph are sufficiently far apart, the precoloring extends to an (r+s)-coloring of G. We determine in all cases the best possible distance bounds between precolored components that allow for this extension result. Next suppose that G is K"r"+"1-minor-free for r=2,3,4, or 5 (and so is r-colorable by proven cases of Hadwiger's Conjecture). Then similarly to the first result there is an extension of a precoloring of a subgraph, each component of which receives s colors from a set of r+s-1, to an (r+s-1)-coloring of the whole graph; we determine bounds on the distance between precolored components that ensures this extension. We include some open problems in this area, building on our joint work with M.O. Albertson.