597 research outputs found
Coloring and covering problems on graphs
The \emph{separation dimension} of a graph , written , is the minimum number of linear orderings of such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} , which is the minimum of such that some linear orderings (repetition allowed) separate every two nonincident edges at least times.
In contrast to separation dimension, we show fractional separation dimension is bounded: always , with equality if and only if contains . There is no stronger bound even for bipartite graphs, since . We also compute for cycles and some complete tripartite graphs. We show that when is a tree and present a sequence of trees on which the value tends to . We conjecture that when the -free -vertex graph maximizing is .
We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let be the number of circular orderings needed to separate all pairs, and let be the fractional version. Among our results: (1) if and only is outerplanar. (2) when is bipartite. (3) . (4) , with equality if and only if . (5) .
A \emph{star -coloring} is a proper -coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring.
A proper vertex coloring of a graph is \emph{-dynamic} if for each , at least colors appear in . We investigate -dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs.
Given a proper total -coloring of a graph , we define the \emph{sum value} of a vertex to be . The smallest integer such that has a proper total -coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} . Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that for any simple graph with maximum degree . We prove this bound to be asymptotically correct by showing that . The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring
On the neighbour sum distinguishing index of planar graphs
Let be a proper edge colouring of a graph with integers
. Then , while by Vizing's theorem, no more than
is necessary for constructing such . On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger , i.e., enables enforcing additional
strong feature of , namely that it attributes distinct sums of incident
colours to adjacent vertices in if only this graph has no isolated edges
and is not isomorphic to . We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph of maximum degree at least which
contains no isolated edges admits a proper edge colouring
such that for every edge of .Comment: 22 page
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