911 research outputs found
A note on 3-colorable plane graphs without 5- and 7-cycles
Borodin et al figured out a gap of the paper published at J. Combinatorial
Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar
technique. The purpose of this note is to fix the gap by slightly revising the
definition of special faces, and adding a few lines of explanation in the
proofs (new added text are all in black font).Comment: 6 page
Characterization of cycle obstruction sets for improper coloring planar graphs
For nonnegative integers , a graph is -colorable if its vertex set can be partitioned into parts so that the
th part induces a graph with maximum degree at most for all . A class of graphs is {\it balanced
-partitionable} and {\it unbalanced -partitionable} if there exists a
nonnegative integer such that all graphs in are -colorable and -colorable, respectively, where the tuple
has length .
A set of cycles is a {\it cycle obstruction set} of a class
of planar graphs if every planar graph containing none of the cycles in as
a subgraph belongs to . This paper characterizes all cycle
obstruction sets of planar graphs to be balanced -partitionable and
unbalanced -partitionable for all ; namely, we identify all
inclusion-wise minimal cycle obstruction sets for all .Comment: 24 pages, 9 figure
Plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable
Listed as No. 53 among the one hundred famous unsolved problems in [J. A.
Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008] is Steinberg's
conjecture, which states that every planar graph without 4- and 5-cycles is
3-colorable. In this paper, we show that plane graphs without 4- and 5-cycles
are 3-colorable if they have no ext-triangular 7-cycles. This implies that (1)
planar graphs without 4-, 5-, 7-cycles are 3-colorable, and (2) planar graphs
without 4-, 5-, 8-cycles are 3-colorable, which cover a number of known results
in the literature motivated by Steinberg's conjecture.Comment: 15 pages, 2 figure. arXiv admin note: text overlap with
arXiv:1506.0462
Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8
We introduce a new variant of graph coloring called correspondence coloring
which generalizes list coloring and allows for reductions previously only
possible for ordinary coloring. Using this tool, we prove that excluding cycles
of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph,
thus answering a question of Borodin.Comment: 22 pages, 3 figures; v2: improves presentatio
Generalization of some results on list coloring and DP-coloring
In this work, we introduce DPG-coloring using the concepts of DP-coloring and
variable degeneracy to modify the proofs on the following papers: (i)
DP-3-coloring of planar graphs without , -cycles and cycles of two
lengths from (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs
and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without
-cycles adjacent simultaneously to -cycles and -cycles is
DP--colorable when (P. Sittitrai, K. Nakprasit,
arXiv:1801.06760(2019) preprint), (iii) Every planar graph is -choosable (C.
Thomassen, J. Combin. Theory Ser. B 62 (1994) 180-181). Using this
modification, we obtain more results on list coloring, DP-coloring,
list-forested coloring, and variable degeneracy.Comment: arXiv admin note: text overlap with arXiv:1807.0081
The proof of Steinberg's three coloring conjecture
The well-known Steinberg's conjecture asserts that any planar graph without
4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic
proof of this conjecture based on the spiral chains of planar graphs proposed
in the proof of the four color theorem by the author in 2004.Comment: 6 pages, 2 figure
List-coloring embedded graphs
For any fixed surface Sigma of genus g, we give an algorithm to decide
whether a graph G of girth at least five embedded in Sigma is colorable from an
assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow
a subgraph (of any size) with at most s components to be precolored, at the
expense of increasing the time complexity of the algorithm to
O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the
multiplicative constant hidden in the O-notation depends on g and s. This also
enables us to find such a coloring when it exists. The idea of the algorithm
can be applied to other similar problems, e.g., 5-list-coloring of graphs on
surfaces.Comment: 14 pages, 0 figures, accepted to SODA'1
Steinberg's Conjecture is false
Steinberg conjectured in 1976 that every planar graph with no cycles of
length four or five is 3-colorable. We disprove this conjecture.Comment: Several typos fixe
Every planar graph without adjacent short cycles is 3-colorable
Two cycles are {\em adjacent} if they have an edge in common. Suppose that
is a planar graph, for any two adjacent cycles and , we have
, in particular, when , . We show that the graph is 3-colorable.Comment: 14 pages, 16 figure
A relaxation of the strong Bordeaux Conjecture
Let be non-negative integers. A graph is
-colorable if the vertex set can be partitioned into
sets , such that the subgraph , induced by
, has maximum degree at most for . Let
denote the family of plane graphs with neither adjacent 3-cycles
nor -cycle. Borodin and Raspaud (2003) conjectured that each graph in
is -colorable. In this paper, we prove that each graph
in is -colorable, which improves the results by Xu
(2009) and Liu-Li-Yu (2014+).Comment: 14 page
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