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It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

Kim gave a new proof of Siegel's Theorem that there are only finitely many S-integral points on \mathbb {P}^1_\mathbb {Z}\setminus \{0,1,\infty \}. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of S increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where S has size 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim. [ABSTRACT FROM AUTHOR]

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ ≥ ln 25 ⁡ Δ , the following holds: for every assignment L of lists of colours to the edges of G , such that | L (e) | ≥ (1 + o (1)) ⋅ max ⁡ { deg ⁡ (u) , deg ⁡ (v) } for each edge e = u v , there is an L -edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k -uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results. [ABSTRACT FROM AUTHOR]

Motivated by the study of greedy algorithms for graph coloring, Bernshteyn and Lee introduced a generalization of graph degeneracy, which is called weak degeneracy. In this paper, we show the lower bound of the weak degeneracy for d -regular graphs is exactly ⌊ d / 2 ⌋ + 1 , which is tight. This result refutes the conjecture of Bernshteyn and Lee on this lower bound. [ABSTRACT FROM AUTHOR]

An adjacent q -cycle is a natural generalization of an adjacent transposition. We show that the number of adjacent q -cycles in a permutation maps to the sum of occurrences of two mesh patterns under Foata's fundamental transformation. As a corollary we resolve Conjecture 3.14 in the paper "From Hertzprung's problem to pattern-rewriting systems" by the first author. [ABSTRACT FROM AUTHOR]

Given a multigraph G and a function f : V (G) → Z ≥ 2 on its vertices, a degree- f subgraph of G is a spanning subgraph in which every vertex v has degree at most f (v). The degree- f arboricity a f (G) of G is the minimum number of colors required to edge-color G into degree- f forests. For every constant function f and for every multigraph G , Truszczyński conjectured that a f (G) ≤ max ⁡ { Δ f (G) + 1 , a (G) } , where Δ f (G) = max v ∈ V (G) ⁡ ⌈ d (v) / f (v) ⌉ and a (G) is the usual arboricity of G. This is a strong generalization of the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary. In this paper, we disprove Truszczyński's conjecture in a strong sense for general multigraphs. On the other hand, extending known results for linear arboricity, we prove that the conjecture holds for simple graphs with sufficiently large girth, and that it holds for all simple graphs asymptotically. More strongly, we prove these partial results in the setting of directed graphs, where the color classes are required to be analogously defined degree- f branchings. [ABSTRACT FROM AUTHOR]

For a digraph G and v ∈ V (G) , let δ + (v) be the number of out-neighbors of v in G. The Caccetta-Häggkvist conjecture states that for all k ≥ 1 , if G is a digraph with n = | V (G) | such that δ + (v) ≥ k for all v ∈ V (G) , then G contains a directed cycle of length at most ⌈ n / k ⌉. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size at least k , has a rainbow cycle of length at most ⌈ n / k ⌉. Let us call (α , β) triangular if every simple edge-colored graph on n vertices with at least αn color classes, each with at least βn edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: • (9 / 8 , 1 / 3) is triangular; • (1 , 2 / 5) is triangular. In this paper, we improve those bounds, showing the following: • (1.1077 , 1 / 3) is triangular; • (1 , 0.3988) is triangular. Our methods give results for infinitely many pairs (α , β) , including β < 1 / 3 ; we show that (1.3481 , 1 / 4) is triangular. [ABSTRACT FROM AUTHOR]