123,826 research outputs found
Parameterized Algorithms for Load Coloring Problem
One way to state the Load Coloring Problem (LCP) is as follows. Let
be graph and let be a 2-coloring. An
edge is called red (blue) if both end-vertices of are red (blue).
For a 2-coloring , let and be the number of red and blue edges
and let . Let be the maximum of
over all 2-colorings.
We introduce the parameterized problem -LCP of deciding whether , where is the parameter. We prove that this problem admits a kernel with
at most . Ahuja et al. (2007) proved that one can find an optimal
2-coloring on trees in polynomial time. We generalize this by showing that an
optimal 2-coloring on graphs with tree decomposition of width can be found
in time . We also show that either is a Yes-instance of -LCP
or the treewidth of is at most . Thus, -LCP can be solved in time
$O^*(4^k).
H-coloring Tori
For graphs G and H, an H-coloring of G is a function from the vertices of G to the vertices of H that preserves adjacency. H-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of H-colorings of the even discrete torus View the MathML source, the graph on vertex set {0,…,m−1}d (m even) with two strings adjacent if they differ by 1 (mod m) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes E and O. In the case m=2 the even discrete torus is the discrete hypercube or Hamming cube Qd, the usual nearest neighbor graph on {0,1}d. We obtain, for any H and fixed m, a structural characterization of the space of H-colorings of View the MathML source. We show that it may be partitioned into an exceptional subset of negligible size (as d grows) and a collection of subsets indexed by certain pairs (A,B)∈V(H)2, with each H-coloring in the subset indexed by (A,B) having all but a vanishing proportion of vertices from E mapped to vertices from A, and all but a vanishing proportion of vertices from O mapped to vertices from B. This implies a long-range correlation phenomenon for uniformly chosen H-colorings of View the MathML source with m fixed and d growing. The special pairs (A,B)∈V(H)2 are characterized by every vertex in A being adjacent to every vertex in B, and having |A||B| maximal subject to this condition. Our main technical result is an upper bound on the probability, for an arbitrary edge uv of View the MathML source, that in a uniformly chosen H-coloring f of View the MathML source the pair View the MathML source is not one of these special pairs (where N⋅ indicates neighborhood). Our proof proceeds through an analysis of the entropy of f, and extends an approach of Kahn, who had considered the case of m=2 and H a doubly infinite path. All our results generalize to a natural weighted model of H-colorings
Ramsey Problems for Berge Hypergraphs
For a graph G, a hypergraph is a Berge copy of G (or a Berge-G in short) if there is a bijection such that for each we have . We denote the family of r-uniform hypergraphs that are Berge copies of G by . For families of r-uniform hypergraphs and , we denote by the smallest number n such that in any red-blue coloring of the (hyper)edges of (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in or a monochromatic red copy of a hypergraph in . denotes the smallest number n such that in any coloring of the hyperedges of with c colors, there is a monochromatic copy of a hypergraph in . In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if , then . In the case r = 2c, we show that , and if G is a noncomplete graph on n vertices, then , assuming n is large enough. In the case we also obtain bounds on . Moreover, we also determine the exact value of for every pair of trees T_1 and T_2.
Read More: https://epubs.siam.org/doi/abs/10.1137/18M1225227?journalCode=sjdme
On Hypergraph Supports
Let be a hypergraph. A support is a graph
on such that for each , the subgraph of induced on the
elements in is connected. In this paper, we consider hypergraphs defined on
a host graph. Given a graph , with ,
and a collection of connected subgraphs of , a primal support
is a graph on such that for each , the
induced subgraph on vertices is connected. A \emph{dual support} is a graph on
s.t. for each , the induced subgraph
is connected, where . We present
sufficient conditions on the host graph and hyperedges so that the resulting
support comes from a restricted family.
We primarily study two classes of graphs: If the host graph has genus
and the hypergraphs satisfy a topological condition of being
\emph{cross-free}, then there is a primal and a dual support of genus at most
. If the host graph has treewidth and the hyperedges satisfy a
combinatorial condition of being \emph{non-piercing}, then there exist primal
and dual supports of treewidth . We show that this exponential blow-up
is sometimes necessary. As an intermediate case, we also study the case when
the host graph is outerplanar. Finally, we show applications of our results to
packing and covering, and coloring problems on geometric hypergraphs
Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree
For graphs , and , write if each
red-blue-edge-coloring of yields a red or a blue . The Ramsey
number is the minimum number such that the complete graph
. In [Discrete Math. 312(2012)], Schelp formulated
the following question: for which graphs there is a constant such
that for any graph of order at least with ,
. In this paper, we prove that for any , if is a
balanced bipartite graph of order with
, then , where
is a matching with edges contained in a connected component. By
Szem\'{e}redi's Regularity Lemma, using a similar idea as introduced by [J.
Combin. Theory Ser. B 75(1999)], we show that for every , there is an
integer such that for any the following holds: Let
such that . Let be a
balanced bipartite graph on vertices with
. Then for each red-blue-edge-coloring
of , either there exist red even cycles of each length in , or there exist blue even cycles of each length
in . Furthermore, the bound
is asymptotically tight. Previous
studies on Schelp's question on cycles are on diagonal case, we obtain an
asymptotic result of Schelp's question for all non-diagonal cases
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