107 research outputs found
DP-4-colorability of two classes of planar graphs
DP-coloring (also known as correspondence coloring) is a generalization of
list coloring introduced recently by Dvo\v{r}\'ak and Postle (2017). In this
paper, we prove that every planar graph without -cycles adjacent to
-cycles is DP--colorable for and . As a consequence, we obtain
two new classes of -choosable planar graphs. We use identification of
verticec in the proof, and actually prove stronger statements that every
pre-coloring of some short cycles can be extended to the whole graph.Comment: 12 page
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
Planar graphs without normally adjacent short cycles
Let be the class of plane graphs without triangles normally
adjacent to -cycles, without -cycles normally adjacent to
-cycles, and without normally adjacent -cycles. In this paper, it is
showed that every graph in is -choosable. Instead of proving
this result, we directly prove a stronger result in the form of "weakly"
DP--coloring. The main theorem improves the results in [J. Combin. Theory
Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently,
every planar graph without -, -, -cycles is -choosable, and every
planar graph without -, -, -, -cycles is -choosable. In the
third section, it is proved that the vertex set of every graph in
can be partitioned into an independent set and a set that induces a forest,
which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In
the final section, tightness is considered.Comment: 19 pages, 3 figures. The result is strengthened, and a new result is
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Some orientation theorems for restricted DP-colorings of graphs
We define signable, generalized signable, and -signable correspondence
assignments on multigraphs, which generalize good correspondence assignments as
introduced by Kaul and Mudrock. DP-colorings from these classes generalize
signed colorings, signed -colorings, and signed list colorings of
signed graphs. We introduce an auxiliary digraph that allows us to prove an
Alon-Tarsi style theorem for DP-colorings from -signable correspondence
assignments on multigraphs, and obtain three DP-coloring analogs of the
Alon-Tarsi theorem as corollaries.Comment: 15 page
On the minimum and maximum selective graph coloring problems in some graph classes
Given a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in Demange et al. (2015). Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivat
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