graph coloring

(redirected from Vertex colouring)

graph coloring

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If an edge k-weighting gives rise to such a proper vertex colouring, we say that the weighting is a vertex colouring by sums.
One may also obtain a vertex colouring from an edge k-weighting by considering the products, sets, or multisets of incident edge weights.
The smallest k for which a graph G has a total k-weighting which is a proper vertex colouring by sums, products, sets or multisets is denoted [X.
The following conjecture motivates the study of total weightings and vertex colouring by sums:
Given a graph G, the smallest k such that any assignment of lists of size k to E(G) permits an edge k-list-weighting which is a vertex colouring by sums is denoted [ch.
An S-edge-weighting gives a vertex colouring if the weighted degrees of adjacent vertices are different.
Thus if one choice of a and b gives a vertex colouring, then any other choice of a and b will as well.
4]) = b does not yield a proper vertex colouring of G if and only if either w([v.
The weighting given in each case gives a proper vertex colouring.
Each of these edge-weightings gives a proper vertex colouring.
The k-total-weighting is neighbour-distinguishing (or vertex colouring, see (1)) if for every edge uv, w(u) + [[SIGMA].
Informally speaking, a track layout of a graph consists of a proper vertex colouring, and a total order of each colour class, such that between each pair of colour classes, no two edges cross (with respect to the orders of the colour classes that contain the endpoints of the edges).