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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.

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

For n≥2 B(K_2,n,Z_2n)≤4n-3 .

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

Let k ≥5 be an odd integer and G be a (n,d,λ) -graph satisfying d^k-1≫ λ^k-2 . Then G has global resilience (1 / 4+o(1)) n d with respect to being C_k -free.

Note that the divisibility conditions in (22) should be equivalent to those in (23) if a t-(n,k,λ) exists. It is open if they are equivalent.

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.