6,649 research outputs found
A generalization of chromatic index
AbstractLet G = (V, E) be a graph and k ⩾ 2 an integer. The general chromatic index χ′k(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph 〈F〉 induced sby F has size at most k - 1. This paper initiates a study of χ′k(G) and generalizes some known results on chromatic index
Edge-colouring graphs with local list sizes
The famous List Colouring Conjecture from the 1970s states that for every
graph the chromatic index of is equal to its list chromatic index. In
1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds
asymptotically. Our main result is a local generalization of Kahn's theorem.
More precisely, we show that, for a graph with sufficiently large maximum
degree and minimum degree , the following
holds: for every assignment of lists of colours to the edges of , such that
for
each edge , there is an -edge-colouring of . Furthermore, Kahn
showed that the List Colouring Conjecture holds asymptotically for linear,
-uniform hypergraphs, and recently Molloy generalized Kahn's original result
to correspondence colouring as well as its hypergraph generalization. We prove
local versions of all of these generalizations by showing a weighted version
that simultaneously implies all of our results.Comment: 22 page
Graph Theory versus Minimum Rank for Index Coding
We obtain novel index coding schemes and show that they provably outperform
all previously known graph theoretic bounds proposed so far. Further, we
establish a rather strong negative result: all known graph theoretic bounds are
within a logarithmic factor from the chromatic number. This is in striking
contrast to minrank since prior work has shown that it can outperform the
chromatic number by a polynomial factor in some cases. The conclusion is that
all known graph theoretic bounds are not much stronger than the chromatic
number.Comment: 8 pages, 2 figures. Submitted to ISIT 201
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