Problems

(i) Is there an integer t such that every t-tough locally finite graph contains a Hamilton circle? (ii) Is there an integer t such that if deleting t k vertices from a locally finite graph G never leaves more than k infinite components…

For every k ≥ 3, we have r(k) = k.

Let G be a graph with maximum degree at most three. Suppose that G has an m-covering by 5-cycles, for some positive integer m. Then G is one of the seven graphs depicted in Figure 7.11.

Does the list of 5-capacities, presented in Section 4, include all members of GI_5 ?

Find an efficient algorithm to find L_{G+H}(\lambda,k)=max\left{L_{G}(\lambda,\ell)\cdots L_{H}(\lambda-\ell,k-\ell)\text{ for some }\chi(G)\leq \ell\leq k-1\right}.

Let G ≅∪_j=1^2KT_j be a disjoint union of an even number of path-like trees, all of them of the same order, and such that T_j≠P_2 for j=1,2, ..., 2 K. Is G a super edge-magic graph?

Let f(G) be the number of matchings of G. Is f(G)^1/|G|≥ f(H)^1/|H| (5.3) when H fractionally tiles G?

There are conjectured recurrences for T(m, n, k) (see A197654), but so far they are unproved.

For even value of m, it seems that rn(G(4mk+2m; 1,2m) = 2mk^2 + 2m^2k + 5mk + m^2 + m - k/2, if k is odd; 2mk^2 + 2m^2k + 7mk + m^2 + 2m - k - 1/2, if k is even.

Investigating Proposition 17, is there a more convenient expression for the upper bound based only on the Young diagram (see Figures 2 and 4) of the set of CRGs K(a, c) : H ↔_c K(a, c), ∀ H ∈ F(H)?

Considering the graphs H_3(d) leads to the conjecture of e(G^3)≥ 2e(G),for G regular, connected, and diam(G) ≥ 3.

For even m and n > m ≥ 4, Δ(T_n) ≤ n - m + 2 ⇒ R(T_n, W_m) = 2n - 1.

A set S of patterns is uncovered if and only if it satisfies f_n = O(t_n+1).

How can we read the combinatorics of the point configuration S ⊂ TT^X from the split system of C_S ?

Is it true that for every nonnegative integer k, there exists a connected hypergraph H satisfying φ_ST(H) − κ_ST(H) + 1 = k?

Let G be a 2-connected bipartite graph with sides A and B satisfying |N^2(X)| ≥ |X| for every X ⊆ A of size at least 3. For every X ⊆ A, |X| ≥ 3, there is a cycle C_X in G such that V(C_X) ∩ A = X.

Let G be a fuzzy graph. Then, (1) s(G) ≤ 2Δ(G) + 1, (2) for every integer k ≥ 2, there exists a fuzzy graph G_k such that k-1 ≤ Δ(G_k) and s(G_k) = Δ(G_k) + k.

First of all, is it possible to sharpen Theorem 1 to the assertion that ‘colex is best’: if A ⊂ [n]^(3), and C is the set of the first |A| elements of [n]^(3) in the colex order, then must we have |VA| ≥ |VC|?

Suppose that G ∈ G^r . Is λ_min^(p)(G) continuously differentiable for p>r ? Is λ_min^(p)(G) continuously differentiable for p \ne k, k=2, ..., r ?

For which graphs H does there exist an H' such that π_G^(H') determines π_G^(H) for every graph G?