4,367 research outputs found
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
Normal edge-colorings of cubic graphs
A normal -edge-coloring of a cubic graph is an edge-coloring with
colors having the additional property that when looking at the set of colors
assigned to any edge and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
the smallest , for which admits a normal
-edge-coloring. Normal -edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with . On the other hand, the known
best general upper bound for was . Here, we improve it by
proving that for any simple cubic graph , which is best
possible. We obtain this result by proving the existence of specific no-where
zero -flows in -edge-connected graphs.Comment: 17 pages, 6 figure
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
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