19 research outputs found
Distance-two coloring of sparse graphs
Consider a graph and, for each vertex , a subset
of neighbors of . A -coloring is a coloring of the
elements of so that vertices appearing together in some receive
pairwise distinct colors. An obvious lower bound for the minimum number of
colors in such a coloring is the maximum size of a set , denoted by
. In this paper we study graph classes for which there is a
function , such that for any graph and any , there is a
-coloring using at most colors. It is proved that if
such a function exists for a class , then can be taken to be a linear
function. It is also shown that such classes are precisely the classes having
bounded star chromatic number. We also investigate the list version and the
clique version of this problem, and relate the existence of functions bounding
those parameters to the recently introduced concepts of classes of bounded
expansion and nowhere-dense classes.Comment: 13 pages - revised versio
Rank-width and Tree-width of H-minor-free Graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r
as a topological minor (resp. as a subgraph) is bounded by a linear (resp.
polynomial) function of their rank-width. We also present refinements of our
bounds for other graph classes such as K_r-minor free graphs and graphs of
bounded genus.Comment: 17 page
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
Modulo-Counting First-Order Logic on Bounded Expansion Classes
We prove that, on bounded expansion classes, every first-order formula with
modulo counting is equivalent, in a linear-time computable monadic lift, to an
existential first-order formula. As a consequence, we derive, on bounded
expansion classes, that first-order transductions with modulo counting have the
same encoding power as existential first-order transductions. Also,
modulo-counting first-order model checking and computation of the size of sets
definable in modulo-counting first-order logic can be achieved in linear time
on bounded expansion classes. As an application, we prove that a class has
structurally bounded expansion if and only if is a class of bounded depth
vertex-minors of graphs in a bounded expansion class. We also show how our
results can be used to implement fast matrix calculus on bounded expansion
matrices over a finite field.Comment: submitted to CSGT2022 special issu
Strong modeling limits of graphs with bounded tree-width
The notion of first order convergence of graphs unifies the notions of
convergence for sparse and dense graphs. Ne\v{s}et\v{r}il and Ossona de Mendez
[J. Symbolic Logic 84 (2019), 452--472] proved that every first order
convergent sequence of graphs from a nowhere-dense class of graphs has a
modeling limit and conjectured the existence of such modeling limits with an
additional property, the strong finitary mass transport principle. The
existence of modeling limits satisfying the strong finitary mass transport
principle was proved for first order convergent sequences of trees by
Ne\v{s}et\v{r}il and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52]
and for first order sequences of graphs with bounded path-width by Gajarsk\'y
et al. [Random Structures Algorithms 50 (2017), 612--635]. We establish the
existence of modeling limits satisfying the strong finitary mass transport
principle for first order convergent sequences of graphs with bounded
tree-width.Comment: arXiv admin note: text overlap with arXiv:1504.0812