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There exists a constant c>1, such that for any triangle-free planar n-vertex graph G,P_DP(G,4)≥ c^n .

We conjecture a hexagon conditional local move as well:

Problem 6.1. Suppose 1/n ≪ε ≪ 1/k . Let X, Y be disjoint sets with |X|=ε n and |Y|=n. Let c=c(n) and t=t(n) with c≤ t ≤|X| . Let H be a k-graph on X ∪ Y with δ_k-1(H)≥ t-c such that H[Y] is independent. What is the complexity of deciding…

Fix d ≥ 2 and ζ > 0. Suppose G is an k-uniform set system on [n], where ζ n < k < n/2, n is sufficiently large, and either G contains no strong d-simplex or G contains no d-cluster. If |G| > (1+ζ)binomn-2k-2, then G is a star.

If k is a power of a prime, P_L_k(G,k)=P_DP(G,k).

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…

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?

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

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?