3,051 research outputs found
Multiple Coloring of Cone Graphs
100學年度研究獎補助論文[[abstract]]A k-fold coloring of a graph assigns to each vertex a set of k colors, and color sets assigned to adjacent vertices are disjoint. The kth chromatic number Xk(G) of a graph G is the minimum total number of colors needed in a k-fold coloring of G. Given a graph G = (V, E) and an integer m ≥ 0, the m-cone of G, denoted by µm(G), has vertex set (V x {0,1,… , m}) U {u} in which u is adjacent to every vertex of V x {m}, and (x, i)(y, j) is an edge if xy ∈ E and i = j = 0 or xy ∈ E and |i - j| = 1. This paper studies the kth chromatic number of the cone graphs. An upper bound for Xk(µm(G) in terms of Xk(G), k, and m are given. In particular, it is proved that for any graph G, if m ≥ 2k, then Xk(µm(G)) ≤ Xk(G) + 1. We also find a surprising connection between the kth chromatic number of the cone graph of G and the circular chromatic number of G. It is proved that if Xk(G)/k > Xc((G) and Xk(G) is even, then for sufficiently large m, Xk(µm(G)) = Xk(G). In particular, if X(G) > Xc(G) and X(G) is even, then for sufficiently large m, X(µm(G)) = X(G).[[notice]]補正完畢[[incitationindex]]SCI[[booktype]]紙
Choosing Colors for Geometric Graphs via Color Space Embeddings
Graph drawing research traditionally focuses on producing geometric
embeddings of graphs satisfying various aesthetic constraints. After the
geometric embedding is specified, there is an additional step that is often
overlooked or ignored: assigning display colors to the graph's vertices. We
study the additional aesthetic criterion of assigning distinct colors to
vertices of a geometric graph so that the colors assigned to adjacent vertices
are as different from one another as possible. We formulate this as a problem
involving perceptual metrics in color space and we develop algorithms for
solving this problem by embedding the graph in color space. We also present an
application of this work to a distributed load-balancing visualization problem.Comment: 12 pages, 4 figures. To appear at 14th Int. Symp. Graph Drawing, 200
The Coloring Ideal and Coloring Complex of a Graph
Let be a simple graph on vertices. We define a monomial ideal in
the Stanley-Reisner ring of the order complex of the Boolean algebra on
atoms. The monomials in are in one-to-one correspondence with the proper
colorings of . In particular, the Hilbert polynomial of equals the
chromatic polynomial of .
The ideal is generated by square-free monomials, so is the
Stanley-Reisner ring of a simplicial complex . The -vector of is a
certain transformation of the tail of the chromatic polynomial
of . The combinatorial structure of the complex is described
explicitly and it is shown that the Euler characteristic of equals the
number of acyclic orientations of .Comment: 13 pages, 3 figure
Graph coloring with no large monochromatic components
For a graph G and an integer t we let mcc_t(G) be the smallest m such that
there exists a coloring of the vertices of G by t colors with no monochromatic
connected subgraph having more than m vertices. Let F be any nontrivial
minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any
n-vertex graph G \in F. This bound is asymptotically optimal and it is attained
for planar graphs. More generally, for every such F and every fixed t we show
that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G
with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}).
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo,
and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5.
We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n),
and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex
graphs of maximum degree 7, average degree at most 6+\epsilon for all
subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only
that the maximum order of magnitude of \mcc_2 is between \sqrt n and n.
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Hyperplanes of the form x_j = x_i + c are called affinographic. For an
affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we
study the function f(M) that counts integral points in [1,M]^n that do not lie
in any hyperplane of the arrangement. We show that f(M) is a piecewise
polynomial function of positive integers M, composed of terms that appear
gradually as M increases. Our approach is to convert the problem to one of
counting integral proper colorations of a rooted integral gain graph. An
application is to interval coloring in which the interval of available colors
for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo
M; the number of proper modular colorations is a different piecewise polynomial
that for large M becomes the characteristic polynomial of the arrangement (by
which means Athanasiadis previously obtained that polynomial). We also study
this function for all positive moduli.Comment: 13 p
Cones of closed alternating walks and trails
Consider a graph whose edges have been colored red and blue. Assign a
nonnegative real weight to every edge so that at every vertex, the sum of the
weights of the incident red edges equals the sum of the weights of the incident
blue edges. The set of all such assignments forms a convex polyhedral cone in
the edge space, called the \emph{alternating cone}. The integral (respectively,
) vectors in the alternating cone are sums of characteristic vectors
of closed alternating walks (respectively, trails). We study the basic
properties of the alternating cone, determine its dimension and extreme rays,
and relate its dimension to the majorization order on degree sequences. We
consider whether the alternating cone has integral vectors in a given box, and
use residual graph techniques to reduce this problem to searching for a closed
alternating trail through a given edge. The latter problem, called alternating
reachability, is solved in a companion paper along with related results.Comment: Minor rephrasing, new pictures, 14 page
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
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