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Can we generalize the property K_n+1∈ K_n,K_n+1 to the Whitney- and r-Whitney numbers?

If G=(V, E) is self-complementary and vertice-transitive,then ρ^⊥(G)=⌈ n/2 ⌉

Let n_1 ≥ n_2 ≥ … ≥ n_t ≥ 4 be positive integers such that at most one of n_2, n_3, …, n_t is odd. Then R(P_n_1, P_n_2, …, P_n_t) = n_1 + ∑_i=2^t (⌊ n_i/2 ⌋ - 1).

Suppose that the graphs Γ_1 and Γ_2 have the same refined spectra. Are their complements cospectral?

Can Theorem 1.3 and Theorem 2.11 be generalized to signed tropical convexity?

χ(2d,d+1)=d+2.

Give a sharp upper bound on γ(X(G)) in terms of γ(G) for any connected graph G with δ(G) ≥ 2, where γ denotes the domination number.

It is natural to conjecture that ker(f^*)=Z_d+2^2⊕ Z_d(d+2)^β(G)-2 holds for non-bipartite graphs in general.

An interesting open question is whether L(p, 1)-LABELING parameterized by only twin cover number is FPT or not.

Let \alpha, \beta and \mu be partitions of the same size. The generating function \sum_{N \ge 0} L_{N\alpha,N\beta}^{N\mu}(q) t^N is a rational function of q and t.

Suppose n = n_1 + … + n_d and k ≥ k_1 + … + k_d, where n_i > k_i ≥ 0 are integers. Let X_1 ∪ … ∪ X_d be a partition of [n] with |X_i| = n_i, and H := F ⊆ binom[n]k : |F ∩ X_i| ≥ k_i for i = 1, …, d . If n_i ≥ 2k_i for all i and n_i > k -…

Charactrize all graphs G which diam(G)=2 and diam(D_{2}(G))=2 or 3.

Do there exist polynomials f_{1}(\zeta),f_{2}(\zeta),f_{3}(\zeta) and f_{4}(\zeta) in \zeta of order 6 , given as in (15), satisfying (16) and (17), such that f_{1}(1)=0,\quad f_{2}(1)^{2}=f_{3}(1)^{2}=f_{4}(1)^{2}=3^{2}?

Let n,r_1,...,r_m∈ Z^+ with r_1+… +r_m≡ 1(bmod 2) and j ∈N , there holds∑_k=0^nη_k∏_i=1^mA_n+i-1,k(q)^r_i≡ 0 bmod 1/[n+1][ c2n n ],where η_k=q^j(k^2+k) or η_k=(-1)^kq^( ck+12 )+j(k^2+k) .

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.