187,748 research outputs found
Light subgraphs in graphs with average degree at most four
A graph is said to be {\em light} in a family of graphs if
at least one member of contains a copy of and there exists
an integer such that each member of
with a copy of also has a copy of such that
for all . In this
paper, we study the light graphs in the class of graphs with small average
degree, including the plane graphs with some restrictions on girth.Comment: 12 pages, 18 figure
Nombre chromatique fractionnaire, degré maximum et maille
We prove new lower bounds on the independence ratio of graphs of maximum degree â â {3,4,5} and girth g â {6,âŠ,12}, notably 1/3 when (â,g)=(4,10) and 2/7 when (â,g)=(5,8). We establish a general upper bound on the fractional chromatic number of triangle-free graphs, which implies that deduced from the fractional version of Reed's bound for triangle-free graphs and improves it as soon as â â„ 17, matching the best asymptotic upper bound known for off-diagonal Ramsey numbers. In particular, the fractional chromatic number of a triangle-free graph of maximum degree â is less than 9.916 if â=17, less than 22.17 if â=50 and less than 249.06 if â=1000. Focusing on smaller values of â, we also demonstrate that every graph of girth at least 7 and maximum degree â has fractional chromatic number at most min (2â + 2^{k-3}+k)/k pour k â â. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree â is at most (2â+9)/5 when â â [3,8], at most (â+7)/3 when â â [8,20], at most (2â+23)/7 when â â [20,48], and at most â/4+5 when â â [48,112]
Sparse halves in dense triangle-free graphs
Erd\H{o}s conjectured that every triangle-free graph on vertices
contains a set of vertices that spans at most
edges. Krivelevich proved the conjecture for graphs with minimum degree at
least . Keevash and Sudakov improved this result to graphs with
average degree at least . We strengthen these results by showing
that the conjecture holds for graphs with minimum degree at least
and for graphs with average degree at least for some absolute . Moreover, we show that the
conjecture is true for graphs which are close to the Petersen graph in edit
distance.Comment: 23 page
The number and average size of connected sets in graphs with degree constraints
The average size of connected vertex subsets of a connected graph generalises a muchâstudied parameter for subtrees of trees. For trees, the possible values of this parameter are critically affected by the presence or absence of vertices of degree 2. We answer two questions of Andrew Vince regarding the effect of degree constraints on general connected graphs. We give a new lower bound, and the first nontrivial upper bound, on the maximum growth rate of the number of connected sets of a cubic graph, and in fact obtain nontrivial upper bounds for any constant bound on the maximum degree. We show that the average connected set density is bounded away from 1 for graphs with no vertex of degree 2, and generalise a classical result of Jamison for trees by showing that in order for the connected set density to approach 1, the proportion of vertices of degree 2 must approach 1. Finally, we show that any sequence of graphs with minimum degree tending to infinity must have connected set density tending to 1/2
Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform
We enumerate the inequivalent self-dual additive codes over GF(4) of
blocklength n, thereby extending the sequence A090899 in The On-Line
Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a
well-known interpretation as quantum codes. They can also be represented by
graphs, where a simple graph operation generates the orbits of equivalent
codes. We highlight the regularity and structure of some graphs that correspond
to codes with high distance. The codes can also be interpreted as quadratic
Boolean functions, where inequivalence takes on a spectral meaning. In this
context we define PAR_IHN, peak-to-average power ratio with respect to the
{I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is
equivalent to the the size of the maximum independent set over the associated
orbit of graphs. Finally we propose a construction technique to generate
Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South
Korea, October 2004. 17 pages, 10 figure
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