Graph Theory: Lecture No. 21
Let d= (d1,d2, . . . , dn)be a sequence of
non-negative integers whose sum is m. We
define the weight of dby
w(d) = Σσ(D)
where the sum is taken over all orientations D
of Gwhose out degree sequence is d.
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Non Negative Integers - Graph Theory - Lecture Slides, Slides of Design Patterns
The key points discuss in the Graph theory, which I found very informative are: Minimum Number, Separating, Disjoint, Edges Separating, Edge Disjoint, Connected, Two Vertices, Inner Vertex, Linked, Vertices
Typology: Slides
2012/2013
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Let d = (d 1 , d 2 ,... , dn) be a sequence of non-negative integers whose sum is m. We define the weight of d by
w (d) = Σσ(D)
where the sum is taken over all orientations D of G whose out degree sequence is d.
Setting xd^ = Πni=1xi di
A(G , X ) = Σd w (d)xd
THE COMBINATORIAL
NULLSTELLENSATZ: Let f be a polynomial over a field F in the variables x = (x 1 , x 2 ,... , xn). Suppose that the total degree of f is Σni=1di and that the coefficients in f of Πxi di^ non-zero. Let Li be a set of di + 1 elements of F , 1 ≤ i ≤ n. Then there exists a t ∈ L 1 ×... × Ln such that f (t) 6 = 0.
If G has an odd number of orientations D with outdegree sequence d. Then G is d + 1 list colourable.
Let G be a graph and let D be an orientation of G without directed odd cycles. Then G is (d + 1)-list colorable, where d is the outdegree sequence of G.
C (G , k) = C (G − e, k) − C (G /e, k)
Colouring of Digraphs: Gallai-Roy Theorem: Every digraph D contains a directed path with χ vertices.
Critical Graphs: A graph G is colour critical if χ(H) < χ(G ) for every proper subgraph H of G. A k-critical graph is one that is k-chromatic and critical.
No critical graph has a clique cut.
Every critical graph is non-separable.