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ABSTRACT If is a list assignment of colors to each vertex of an ‐vertex graph , then anequitable‐coloringof is a proper coloring of vertices of from their lists such that no color is used more than times. A graph isequitably‐choosableif it has an equitable ‐coloring for every ‐list assignment . In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer every graph with maximum degree at most is equitably ‐choosable. The main result of this paper is that for each and each planar graph , a stronger statement holds: if the maximum degree of is at most , then is equitably ‐choosable. In fact, we prove the result for a broader class of graphs—the class of the graphs in which each bipartite subgraph with has at most edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in , in particular, for all planar graphs. We also introduce the new stronger notion ofstrongly equitable(SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE ‐choosable, then it is both equitably ‐choosable and equitably ‐colorable, while neither of being equitably ‐choosable and equitably ‐colorable implies the other.
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Decreasing the mean subtree order by adding k edges
Abstract Themean subtree orderof a given graph , denoted , is the average number of vertices in a subtree of . Let be a connected graph. Chin et al. conjectured that if is a proper spanning supergraph of , then . Cameron and Mol disproved this conjecture by showing that there are infinitely many pairs of graphs and with , and such that . They also conjectured that for every positive integer , there exists a pair of graphs and with , , and such that . Furthermore, they proposed that provided . In this note, we confirm these two conjectures.
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- Award ID(s):
- 2154331
- PAR ID:
- 10505987
- Publisher / Repository:
- John Wiley
- Date Published:
- Journal Name:
- Journal of Graph Theory
- Volume:
- 105
- Issue:
- 3
- ISSN:
- 0364-9024
- Page Range / eLocation ID:
- 357 to 366
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/jgt.23043
