Capítol V. El Ple
Article 64. Funcionament
The catalog allows for much expansion. In particular, a classification of all good 6-wheels with f (u) = 1 and f (u) = 5 is currently near completion. What other small graphs can be determined to be good? How can we incorporate information obtained about graphs with respect to sum-list-coloring to simplify some of the existing results and make it easier to find new results? For example, knowing the sum choice number of a graph will provide a lower bound for the sum of list sizes needed for a graph to be good. Consider the 4-wheel. It was shown in Chapter6 that χSC(W4) = 12. Thus, the sum of list sizes assigned to the vertices of W4 would need to be at least 12 in order for it to possibly be good.
7.4 Sum-list-coloring
The notion of sum-list-coloring is still a fairly new area and there are not a lot of results known. This means that there are many open questions in the area. Some goals for the near future are to complete a characterization of the sum choice number of all graphs on six vertices and other graphs on a small number of vertices. In particular determining whether the wheels W5, . . . , W10 would complete a classification of all sc-greedy wheels and broken wheels begun by Heinold [35]. While in the midst of working on this characterization, we are exploring the
question of how many colors are needed to create an f -assignment for a graph G that shows f is not a choice function for G. How does this number relate to certain parameters of the graph?
As we showed that paths of cycles and certain trees of cycles are sc-greedy, it is a natural progression to explore showing that cycles of cycles are sc-greedy and that paths of cliques are sc-greedy. In what other ways can we ‘glue’ sc-greedy graphs together to obtain graphs that are sc-greedy? In particular, what if two sc-greedy graphs are joined by a cut edge? These are things we are currently working on.
While it is known which theta graphs are sc-greedy, we seek to determine the sum choice number of generalized theta graphs with an arbitrary number of disjoint paths between two vertices. We note that there are some partial results on this question that I have completed with Michael Young.
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