61 research outputs found
Open problems on graph coloring for special graph classes.
For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:Vâ{1,2,âŠ}c:Vâ{1,2,âŠ} such that c(u)â c(v)c(u)â c(v) for every edge uvâEuvâE. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring
Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget
In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [BartĆomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented.
As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs . We show that if there is a set of non-intersecting unit arcs of size such that does not contain L+1 arcs intersecting in one point, then it is possible to color with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color with L+1 colors or more
Flexible List Colorings in Graphs with Special Degeneracy Conditions
For a given , we say that a graph is
-flexibly -choosable if the following holds: for any assignment
of color lists of size on , if a preferred color from a list is
requested at any set of vertices, then at least of these
requests are satisfied by some -coloring. We consider the question of
flexible choosability in several graph classes with certain degeneracy
conditions. We characterize the graphs of maximum degree that are
-flexibly -choosable for some , which answers a question of Dvo\v{r}\'ak, Norin, and
Postle [List coloring with requests, JGT 2019]. In particular, we show that for
any , any graph of maximum degree that is not isomorphic
to is -flexibly -choosable. Our
fraction of is within a constant factor of being the best
possible. We also show that graphs of treewidth are -flexibly
-choosable, answering a question of Choi et al.~[arXiv 2020], and we give
conditions for list assignments by which graphs of treewidth are
-flexibly -choosable. We show furthermore that graphs of
treedepth are -flexibly -choosable. Finally, we introduce a
notion of flexible degeneracy, which strengthens flexible choosability, and we
show that apart from a well-understood class of exceptions, 3-connected
non-regular graphs of maximum degree are flexibly -degenerate.Comment: 21 pages, 5 figure
Maximizing Happiness in Graphs of Bounded Clique-Width
Clique-width is one of the most important parameters that describes
structural complexity of a graph. Probably, only treewidth is more studied
graph width parameter. In this paper we study how clique-width influences the
complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE)
problems. We answer a question of Choudhari and Reddy '18 about
parameterization by the distance to threshold graphs by showing that MHE is
NP-complete on threshold graphs. Hence, it is not even in XP when parameterized
by clique-width, since threshold graphs have clique-width at most two. As a
complement for this result we provide a algorithm for MHE, where is the number of colors
and is the clique-width of the input graph. We also
construct an FPT algorithm for MHV with running time
, where is the
number of colors in the input. Additionally, we show
algorithm for MHV on interval graphs.Comment: Accepted to LATIN 202
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