116,208 research outputs found
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
A lower bound on independence in terms of degrees
We prove a new lower bound on the independence number of a simple connected graph
in terms of its degrees
On Selkow's bound on the independence number of graphs
For a graph G with vertex set V (G) and independence number α(G), Selkow [A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365] established the famous lower bound ∑v∈V(G)1d(v)+1(1+max{d(v)d(v)+1-∑u∈N(v)1d(u)+1,0}) on α (G), where N(v) and d(v) = |N(v)| denote the neighborhood and the degree of a vertex v ∈ V (G), respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here
Relative Fractional Independence Number and Its Applications
We define the relative fractional independence number of two graphs, and
, as where the maximum is taken over all graphs , is the
strong product of and , and denotes the independence number. We
give a non-trivial linear program to compute and discuss some
of its properties. We show that where
can be the independence number, the zero-error Shannon capacity, the
fractional independence number, the Lov'{a}sz number, or the Schrijver's or
Szegedy's variants of the Lov'{a}sz number of a graph . This inequality is
the first explicit non-trivial upper bound on the ratio of the invariants of
two arbitrary graphs, as mentioned earlier, which can also be used to obtain
upper or lower bounds for these invariants. As explicit applications, we
present new upper bounds for the ratio of the zero-error Shannon capacity of
two Cayley graphs and compute new lower bounds on the Shannon capacity of
certain Johnson graphs (yielding the exact value of their Haemers number).
Moreover, we show that the relative fractional independence number can be used
to present a stronger version of the well-known No-Homomorphism Lemma. The
No-Homomorphism Lemma is widely used to show the non-existence of a
homomorphism between two graphs and is also used to give an upper bound on the
independence number of a graph. Our extension of the No-Homomorphism Lemma is
computationally more accessible than its original version
Nombre chromatique fractionnaire, degré maximum et maille
We prove new lower bounds on the independence ratio of graphs of maximum degree ∆ ∈ {3,4,5} and girth g ∈ {6,…,12}, notably 1/3 when (∆,g)=(4,10) and 2/7 when (∆,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 ∆ ≥ 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 ∆ is less than 9.916 if ∆=17, less than 22.17 if ∆=50 and less than 249.06 if ∆=1000. Focusing on smaller values of ∆, we also demonstrate that every graph of girth at least 7 and maximum degree ∆ has fractional chromatic number at most min (2∆ + 2^{k-3}+k)/k pour k ∈ ℕ. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree ∆ is at most (2∆+9)/5 when ∆ ∈ [3,8], at most (∆+7)/3 when ∆ ∈ [8,20], at most (2∆+23)/7 when ∆ ∈ [20,48], and at most ∆/4+5 when ∆ ∈ [48,112]
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