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Every 3-connected α-tough graph G contains an edge e such that both G - e and G/e are α-tough.

Let k,d,c ∈N^* be parameters, and let F_1,...,F_k be finite sets of irreducible polynomials of degree at most d such that - ∩_iF_i=∅, for every Q_1,...,Q_k-1 each from a distinct set F_i_j , there are polynomials P_1,...,P_c in the…

For any c < 4, there exists a finite list of graphs L such that if G is a critical graph with Ad(G) ≤ c then G ∈ L.

Conjecture 6.2. The generating function H_{m,a}^{(k)}(x) is rational.

Zaleski [30, Conjecture 3.4] conjectured that the distribution of (n, dn-1)-core partitions with distinct parts is asymptotically normal as n tends to infinity when d is given.

For integers n, r and a prime p satisfying r < p, we have ex(n, K_r, C_≥ p^prime) ≤ n-1/p-2 binomp-1r. Equality holds only for connected n-vertex graphs consisting of n-1/p-2 maximal 2-connected blocks each isomorphic to K_p-1.

It meets ρ^⊥(G)>ρ^⊥(V_c) if it meets the following: - ρ^⊥(V_c)>ρ^⊥(A) - ρ^⊥(V_c)>ρ^⊥(B) - ρ^⊥(V_c)≤ ρ^⊥(A)+ρ^⊥(B)

Let n = ks + ℓ where 0 < ℓ < k and F_0, F_1, …, F_s ⊂ binom[n]k be non-empty cross-union families. Does the following inequality hold? ∑_i=0^s |F_i| ≤ max (s+1) binomn-1k, 1 + s binomnk - ∑_i=0^k-ℓ binomki binomn-kk-i

So, we have an open problem, χ(G(2,11,9)=? .

If G is a graph where every precoloring of at most k edges can be extended to a proper χ'(G)-edge coloring, then every precoloring of at most k+1 edges of G square K_2 is extendable to a proper (χ'(G)+1)-edge coloring of G square K_2.

If s > t ≥ 1 with (s, -t) ≠ (2, -1) and n, r ∈ P then ⌊ ( ∑{k=n}^{∞} 1/{rk}{s,-t} )^{-1} ⌋ = {rn}{s,-t} - {r(n-1)}{s,-t} - 1. If t = -1 and s, n, r ∈ P then ⌊ ( ∑{k=n}^{∞} 1/{rk}{s,-1}^2 )^{-1} ⌋ = {rn}{s,-1}^2 - {r(n-1)}{s,-1}^2 - 1.

Let d: V → Z_0^+ be a symmetric function that satisfies d(∅) = 0 and ∀ X, Y ⊆ V (19) and (20). Let hatR: V → Z_0^+ be an even valued, symmetric, skew-supermodular function. Suppose that hatR(X) ≤ d(X) ∀ X ⊆ V. Then there exists a pairing M…

For any sequence s_0,...,s_n of non-negative integers satisfying \sum s_i=\sum i s_i=n, there exist i, j with 1\le i, j and i+j\le n so that n \left( {i+j \atop i} \right)s_{i+j}\geqslant e s_i s_j.

An interesting question is whether every red-blue coloring of a k-pseudorandom graph contains a monochromatic path of length \Omega(\frac{n}{\sqrt{k}}).

If T is a k-peripheral tree and G is a nontrivial connected graph, then aw(T square G,k)=k .

For any 1 ≤ α ≤ 6, p_∞^(α) := lim_n → ∞ p_n^(α) exists and is given by: p_∞^(1) = 1/π (6.1) p_∞^(2) = p_∞^(5) = 1/2 - 1/π (6.2) p_∞^(3) = p_∞^(4) = 2/π - 1/2 (6.3) p_∞^(6) = 1 - 3/π (6.4)

If r is an integer that is not a part in either partitions λ or μ , then L_λ+(r),μ+(r)(q)≥ L_λ μ(q).

In particular, what is the value L_{k,S} = \lim_{n \to \infty} b_{k,S}(n)^{1/n}?

It would be interesting to know if there exists a regular/vertex-transitive self-complementary graph Γ on n vertices with the second eigenvalue in the bounds frac√n(n-4)-12 < λ_2 ≤ n-7/2 - 2cos(π(n-1)/n).

We make the analogous conjecture here, that we may have both t and d in Theorem3.2, provided deg(d)=3.