Problems
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Let G be a tight graph such that: For every edge (u,v)∈ E(G) , one of its endpoints is dense, and the other is non-dense, and |N(u)∩ N(v)|≤ 1 , for all pair of vertices u,v ∈ D(G),u≠v .Then, χ_b(G) = m(G).
In particular, what is the dimension of the restriction of that space to patterns of size k? Is it spanned by corner trees with k vertices?
and for r ≥3 ∑_i=0^n(-1)^n-i( ln i )U_m+k+2+i,k+i^(r)= ll2^n if m=2n, 2^n(5/2n+2r-1) if m=2n+1, . (1.11)
The double sequence A^σ,id assigned to the D'Arcais polynomials is horizontally log-concave.
By using the relation α(G)=ω(Ḡ), can we obtain min(α, r^n), max(α, r^n), Min(α, r^n) and Max(α, r^n)?
Conjecture 5.4. Let w(x;t)=∑_λω(λ)P_λ(x;t), where P_λ(x;t) denote the Hall-Littlewood function corresponding to the partition λ , and the sum runs over all partitions λ . Then rlog w(x;-1)+∑_n ≥ 1 odd1/2na^nc^np_2n+∑_n ≥ 2…
It seems likely that this is the only such case, though we do not have a proof yet.
More general: does it hold for all monadic-second order definable problems? We conjecture not
rgirth(F)≤2 ∑_1≤i≤nfrac1|F_i|+1 for any family F=(F_1,...,F_n) of subsets of E(K_n) .
Let h(t) be a polynomial of degree s ≤ d with nonnegative coefficients that satisfies Properties (H) and (S). Does U_r^d+1h(t) have an interlacing symmetric decomposition for all r ≥ d+1 (possibly even for r ≥ maxs, d+1-s)?
In this range the best lattice for Q3 is also a best lattice for Q2, and we conjecture that this is always true.
The ratio generating A130296.
We conjecture that the range of k ∈ N in Lemma 2.6 can be extended to k ∈ R.
Determine the least positive integer k such that β(G)-β_b(G)≤k for any graph G.
We have [x^n]g_m(x)=[x^n](frac-x(1-W(-x))^mW(-x))^n, where g_m(x) is the generating function of the central coefficients (m-2)n+1/(m-1)n+1( cmn ) of the Riordan array (c(x),xc(x)^m-2) .
When is it that χ_{td}(cl(G))=2 χ_{td}(G)+1 ? Are there infinitely many graphs with this property?
The coefficient of q in hatE_k,n(q) is binomnk+1binomnk-2.
they conjectured that C_m × P_n is SEAT if m ≥ 4 even, n ≥ 3 and d ∈ 0, 2;
Does there exist a natural filtration {0=\mathcal{F}{0}\subset \mathcal{F}{1}\subset \cdots } on the ring of semi-invariants SI(Q,\beta) such that for a special quiver Q=T_{n,n,n} and a special dimensional vector \beta , see [14], Section…
In the poset of graphs and cc mappings between them, is every non-degenerate interval nonempty? Does every nondegenerate interval contain an infinite antichain? Does every nondegenerate interval contain every countable poset?