List coloring of graphs having cycles of length divisible by a given number

Derived colourings of graphs

“…Note th at the class of round graphs is much richer than those trivially obtained by taking a graph and subdividing each edge into a path of length 4; for example, ©(2,2,2) -K 24 is a round graph which is not obtained in this way. Round graphs are a particular case of a larger class of graphs studied by Akbari et al in [6], where they determine the choosability of graphs whose cycles have length a multiple of some fixed integer k > 3.…”

“…A number of the arguments which follow rely on this notion of a-parity. For instance, in some cases G is shown to be {a, 6 Furthermore, i f ( X , Y ) is the bipariition ofV(G) into independent sets with |Aj even, then the weighting may be chosen so that x is a-odd for every and y is a-even for every y E T .…”

“…By using a class of "gadget" graphs, one can explicitly construct infinite classes of graphs which may not be {a, 6 the following examples of graphs which are not 2 -weight colourable may be con structed.…”

“…An {a, 6 }-weight colouring of K2 □ CV The {a, 6 }-weight colouring given for K2 □ Cf can be extended to an {a, fr}-weight colouring of K2 □ Cg by the replacement operation shown inFigure 6. Note that the right subgraph contains the left one, and thus this operation may be repeated as many times as needed to give an {a, 6 }-weight colouring for any K 2 □ C2k+i{k > 3).F ig u re 6 : Replacement operation for {a,6 }-weight colourings of K2 □ C2k+\We now show that K 2 □ C3 has no {a,6 }-weight colouring for any choice of a and b. Let G = (V,E) have vertex and edge sets as follows: V{G) -{xu x2 ,x3 ,yu y2 ,y3}, E(G) = {xxx2, x2x3, x3xx,yxy2 ,y2y3 ,y3 yx,xxyx, x2y2, x3y3}.…”

“…Now, suppose th at only one of yxy2 and y2y3 is weighted a, say w{yxy2) = b and w{y2yz) = a. If w(y3yx) = a, then c.(xx) = c(yx); if w(y3yx) = b, then c(x3) = c{y3).Hence K2 □ C3 has no {a,6 }-weight colouring. Xi a A partial edge {a, 6 }-weighting of K2 □ C3 A similar (and much longer) analysis can show that no {a, fr}-weight colouring of K2 □ C5 exists.…”