265 research outputs found
On (4,2)-Choosable Graphs
A graph is called -choosable if for any list assignment which
assigns to each vertex a set of permissible colours, there is a
-tuple -colouring of . An -choosable graph is also called
-choosable. In the pioneering paper on list colouring of graphs by
Erd\H{o}s, Rubin and Taylor, -choosable graphs are characterized. Confirming
a special case of a conjecture of Erd\H{o}s--Rubin--Taylor, Tuza and Voigt
proved that -choosable graphs are -choosable for any positive
integer . On the other hand, Voigt proved that if is an odd integer,
then these are the only -choosable graphs; however, when is even,
there are -choosable graphs that are not -choosable. A graph is
called -choosable-critical if it is not -choosable, but all its proper
subgraphs are -choosable. Voigt conjectured that for every positive integer
, all bipartite -choosable-critical graphs are -choosable. In
this paper, we determine which -choosable-critical graphs are
-choosable, refuting Voigt's conjecture in the process. Nevertheless, a
weaker version of the conjecture is true: we prove that there is an even
integer such that for any positive integer , every bipartite
-choosable-critical graph is -choosable. Moving beyond
-choosable-critical graphs, we present an infinite family of
non--choosable-critical graphs which have been shown by computer analysis to
be -choosable. This shows that the family of all -choosable
graphs has rich structure.Comment: 18 pages, 8 figures. Now includes source code in ancillary file
Complexity of choosability with a small palette of colors
A graph is -choosable if, for any choice of lists of colors for
each vertex, there is a list coloring, which is a coloring where each vertex
receives a color from its list. We study complexity issues of choosability of
graphs when the number of colors is limited. We get results which differ
surprisingly from the usual case where is implicit and which extend known
results for the usual case. We also exhibit some classes of graphs (defined by
structural properties of their blocks) which are choosable.Comment: 31 pages, 11 figure
Total weight choosability of d-degenerate graphs
A graph is -choosable if the following holds: For any list
assignment which assigns to each vertex a set of real
numbers, and assigns to each edge a set of real numbers, there
is a total weighting
such that for , and
for every edge . This paper proves the following results: (1) If is a
connected -degenerate graph, and is a prime number, and is either
non-bipartite or has two non-adjacent vertices with , then
is -choosable. As a consequence, every planar graph with no isolated
edges is -choosable, and every connected -degenerate non-bipartite
graph other than is -choosable. (2) If is a prime number,
is an ordering of the vertices of such that each
vertex has back degree , then there is a graph
obtained from by adding at most leaf neighbours to (for
each ) and is -choosable. (3) If is -degenerate and
a prime, then is -choosable. In particular, -degenerate graphs
are -choosable. (4) Every graph is -choosable. In particular, planar graphs are
-choosable, planar bipartite graphs are -choosable.Comment: 16 pages, 1 figures
List edge-colouring and total colouring in graphs of low treewidth
We prove that the list chromatic index of a graph of maximum degree
and treewidth is ; and that the total
chromatic number of a graph of maximum degree and treewidth is . This improves results by Meeks and Scott.Comment: 10 page
A Characterization of -Choosable Graphs
A graph is \emph{-choosable} if given any list assignment with
for each there exists a function such that
and for all , and whenever
vertices and are adjacent . Meng,
Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We
prove their conjecture.Comment: 20 pages, 20 figures; version 3 incorporates reviewer feedback:
completely rewrote Section 5 to correct an error, omitted many tedious
details of showing that certain graphs are not (4,2)-choosable, removed open
question and conjecture; to appear in J. Graph Theor
On the list chromatic index of graphs of tree-width 3 and maximum degree at least 7
Among other results, it is shown that 3-trees are -edge-choosable and
that graphs of tree-width 3 and maximum degree at least 7 are
-edge-choosable
List colouring with a bounded palette
Kr\'al' and Sgall (2005) introduced a refinement of list colouring where
every colour list must be subset to one predetermined palette of colours. We
call this -choosability when the palette is of size at most
and the lists must be of size at least . They showed that, for any integer
, there is an integer , satisfying as
, such that, if a graph is -choosable, then it is
-choosable, and asked if is required to be exponential in . We
demonstrate it must satisfy .
For an integer , if is the least integer such that
a graph is -choosable if it is -choosable, then we more
generally supply a lower bound on , one that is super-polynomial in
if , by relation to an extremal set theoretic
property. By the use of containers, we also give upper bounds on
that improve on earlier bounds if .Comment: 12 pages, 1 figure, 1 table; to appear in Journal of Graph Theor
Total weight choosability for Halin graphs
A proper total weighting of a graph is a mapping which assigns to
each vertex and each edge of a real number as its weight so that for any
edge of , . A -list assignment of is a mapping which
assigns to each vertex a set of permissible weights and to each
edge a set of permissible weights. An -total weighting is a
total weighting with for each .
A graph is called -choosable if for every -list assignment
of , there exists a proper -total weighting. As a strenghtening of
the well-known 1-2-3 conjecture, it was conjectured in [ Wong and Zhu, Total
weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every
graph without isolated edge is -choosable. It is easy to verified this
conjecture for trees, however, to prove it for wheels seemed to be quite
non-trivial. In this paper, we develop some tools and techniques which enable
us to prove this conjecture for generalized Halin graphs
Proportional Choosability of Complete Bipartite Graphs
Proportional choosability is a list analogue of equitable coloring that was
introduced in 2019. The smallest for which a graph is proportionally
-choosable is the proportional choice number of , and it is denoted
. In the first ever paper on proportional choosability, it was
shown that when , . In this note we improve on this result
by showing that . In the process, we
prove some new lower bounds on the proportional choice number of complete
multipartite graphs. We also present several interesting open questions.Comment: 11 page
List Coloring and -monophilic graphs
In 1990, Kostochka and Sidorenko proposed studying the smallest number of
list-colorings of a graph among all assignments of lists of a given size
to its vertices. We say a graph is -monophilic if this number is
minimized when identical -color lists are assigned to all vertices of .
Kostochka and Sidorenko observed that all chordal graphs are -monophilic for
all . Donner (1992) showed that every graph is -monophilic for all
sufficiently large . We prove that all cycles are -monophilic for all
; we give a complete characterization of 2-monophilic graphs (which turns
out to be similar to the characterization of 2-choosable graphs given by Erdos,
Rubin, and Taylor in 1980); and for every we construct a graph that is
-choosable but not -monophilic
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