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Let p ≥ 2 be an integer and G be a nonbipartite graph of order n, with minimum degree δ>2n/(2p+3) . Then G contains a cycle of length l, for each integer l,2p≤l≤δ+1 .

Conjecture 1.6.1. The polynomial f_m(b,q) has the form f_m(b,q)=∑_i=0^binomm2(1-q)^m-y^(i)g_m,i(q)b^i where y(n)=⌊frac√8n+12⌋ and g_m,i(q) are polynomials. Further, with <_k^n> denot ing the Eulerian numbers ^3…

There is an isomorphism of graded S_n-modules R_n,k,r≅ R_n,k,r' .

A natural conjecture would be that for any (0,1)-matrix, the lattice formed by its integral null vectors has a small number of near-shortest vectors.

The case σ = 2 can be described using hypergeometric functions; is there a notion of generalized hypergeometric function that could be applied for larger values of σ?

Determine the compound curling numbers different products of graphs in which one graph is a regular graph.

Interestingly, all the properties of Proposition 2 hold even for negative k, and it seems that for any k the A_k(n) eventually become positive for n sufficiently large, ...

(4) χ(G(2,11,9))=4?

In particular, is it true that if the realization |Γ| of Γ through its direct complex Δ(Γ) is a manifold, then the realization of its partial dual |Γ^S| is also a manifold?

Every graph without isolated vertices admits super edge total local antimagic labeling.

For central arrangements whose underlying matroid is connected, the homotopy type of the complement determines the underlying matroid.

Every triangularly connected P_3 -dominated graph on at least three vertices is vertex pancyclic, with an exception K_1,1,3 .

Let 1 ≤ t ≤ r ≤ binomn2. If n is sufficiently large relative to t and r, then the set B_r(n) is t-EKR.

Is it possible to prove an analogue of Theorem 3.1 for non-uniform hypertrees?

Find a function a(k) such that for every k we have the tight bound γ_krt(G) ≤ a(k) · γ_rk(G).

Let G be a 2k-edge-connected graph with k ≥ 1 and let L(v) ⊆ k, …, d_G(v) such that |L(v)| ≥ ⌈ d_G(v)/2 ⌉ - ⌈ k/2 ⌉ + 2 for every v ∈ V(G). Is it true that G includes a k-edge-connected L-factor?

Consider maximal planar graphs, 3-connected planar graphs or planar 3-trees. For all such graphs G, is there a constant α<1 and a constant k such that Z(G)≤ α n+k ?

For d ≥ 3, is every minimum edge cut of a flag d-polytope or a balanced d-polytope trivial?

(1) χ(G(2,7,5))=4?

Is every wqo class of graphs in fact bqo?