20 research outputs found
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Edge and total choosability of near-outerplanar graphs
It is proved that, if G is a K4-minor-free graph with maximum degree ∆ ≥ 4, then G is totally (∆ + 1)-choosable; that is, if every element (vertex or edge) of G is assigned a list of ∆ + 1 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ch’’(G) = χ’(G) for every graph G, is true for all K4-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all K2,3-minor free graphs and all ( K2 + (K1 U K2))-minor-free graphs
List Edge Colorings of Planar Graphs with 7-cycles Containing at Most Two Chords
In this paper we prove that if G is a planar graph, and each 7-cycle contains at most two chords, then G is edge-k-choosable, where k = max{8, ?(G) + 1}
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
The List Square Coloring Conjecture fails for cubic bipartite graphs and planar line graphs
Kostochka and Woodall (2001) conjectured that the square of every graph has
the same chromatic number and list chromatic number. In 2015 Kim and Park
disproved this conjecture for non-bipartie graphs and alternatively they
developed their construction to bipartite graphs such that one partite set has
maximum degree . Motivated by the List Total Coloring Conjecture, they also
asked whether this number can be pushed down to . At about the same time,
Kim, SooKwon, and Park (2015) asked whether there would exist a claw-free
counterexample to establish a generalization for a conjecture of Gravier and
Maffray (1997). In this note, we answer the problem of Kim and Park by pushing
the desired upper bound down to by introducing a family of cubic bipartite
counterexamples, and positively answer the problem of Kim, SooKwon, and Park by
introducing a family of planar line graphs
Towards on-line Ohba's conjecture
The on-line choice number of a graph is a variation of the choice number
defined through a two person game. It is at least as large as the choice number
for all graphs and is strictly larger for some graphs. In particular, there are
graphs with whose on-line choice numbers are larger
than their chromatic numbers, in contrast to a recently confirmed conjecture of
Ohba that every graph with has its choice number
equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture
was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method
to on-line colouring of graphs, European J. Combin., 2011]: Every graph
with has its on-line choice number equal its chromatic
number. This paper confirms the on-line version of Ohba conjecture for graphs
with independence number at most 3. We also study list colouring of
complete multipartite graphs with all parts of size 3. We prove
that the on-line choice number of is at most , and
present an alternate proof of Kierstead's result that its choice number is
. For general graphs , we prove that if then its on-line choice number equals chromatic number.Comment: new abstract and introductio