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In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

For a compact set A \subset \mathbb {R}^d and an integer k\ge 1, let us denote by \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that \frac {1}{k}A[k] converges to the convex hull of A in Hausdorff distance as k tends to infinity. Bobkov, Madiman and Wang [ Concentration, functional inequalities and isoperimetry , Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of \frac {1}{k}A[k] is nondecreasing in k, or in other words, in terms of the volume deficit between the convex hull of A and \frac {1}{k}A[k], this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if d=1 but fails for any d \geq 12. In this paper we show that the conjecture is true for any star-shaped set A \subset \mathbb {R}^d for d=2 and d=3 and also for arbitrary dimensions d \ge 4 under the condition k \ge (d-1)(d-2). In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in \mathbb {R}^d, for any d \geq 7. [ABSTRACT FROM AUTHOR]

Let r \geq k \geq 2 and K(k) r,n denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A decomposition of K(k) r,n is a partition of E(K(k) r,n). A perfect matching (resp., Hamilton tight cycle) decomposition of K(k) r,n is a decomposition of K(k) r,n into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if k | n (resp., 2 - k and k | n), then K(k) k+1,n (resp., K(k) k+2,n) has a perfect matching decomposition. We also prove that for any integer k \geq 2, K(k) k+1,n has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [Appl. Math. Comput., 386 (2020), 125492]. [ABSTRACT FROM AUTHOR]

Let G be a graph and k an integer with 2 ≤ k ≤ n . The k-tree-connectivity of G, denoted by κ k (G) , is defined as the minimum κ G (S) over all k-subsets S of vertices, where κ G (S) denotes the maximum number of internally disjoint S-trees in G. The k-path-connectivity of G, denoted by π k (G) , is defined as the minimum π G (S) over all k-subsets S of vertices, where π G (S) denotes the maximum number of internally disjoint S-paths in G. In this paper, we first prove that for any fixed integer k ≥ 1 , given a graph G and a subset S of V(G), deciding whether π G (S) ≥ k is NP -complete. Moreover, we also show that for any fixed integer k 1 ≥ 5 , given a graph G, a k 1 -subset S of V(G) and an integer 1 ≤ k 2 ≤ n - 1 , deciding whether π G (S) ≥ k 2 is NP -complete. Let π (k , ℓ) = 1 + max { κ (G) | G \ is\ a\ graph\ with π k (G) < ℓ } . Hager (Discrete Math 59:53–59, 1986) showed that ℓ (k - 1) ≤ π (k , ℓ) ≤ 2 k - 2 ℓ and conjectured that π (k , ℓ) = ℓ (k - 1) for k ≥ 2 and ℓ ≥ 1 . He also confirmed the conjecture for 2 ≤ k ≤ 4 and proved π (5 , ℓ) ≤ ⌈ 9 2 ℓ ⌉ . By introducing a "Generalized Path-Bundle Transformation", we confirm the conjecture for k = 5 and prove that π (k , ℓ) ≤ 2 k - 3 ℓ for k ≥ 5 and ℓ ≥ 1 . By employing this transformation, we also prove that if G is a graph with κ (G) ≥ (k - 1) ℓ for any k ≥ 2 and ℓ ≥ 1 , then κ k (G) ≥ ℓ . [ABSTRACT FROM AUTHOR]

• We disprove a conjecture proposed by Fiedorowicz, Sidorowicz, Sopena (Appl. Math. Comput. 410 (2021) 126253). • In addition, we find a result of the above paper is not correct. • Two sufficient conditions are given for a hamiltonian digraph D with w c → (D) = 2. Under an arc-coloring c of a digraph D , if for each pair of vertices (u , v) , there exists a directed walk from u to v satisfying that any two consecutive arcs of it have different colors, we say that D is properly-walk connected, and c is a proper-walk coloring of D. The proper-walk connection number w c → (D) of D is the least integer k such that D has a proper-walk coloring with k colors. Fiedorowicz, Sidorowicz, Sopena (Appl. Math. Comput. 410 (2021) 126253) conjectured that if D is a hamiltonian digraph with δ (D) ≥ 2 , then w c → (D) ≤ 2. In this paper, we disprove the conjecture by constructing two families of counterexamples. Also, we present some cases for a hamiltonian digraph D having w c → (D) = 2. In addition, we find that Observation 15 is not true in the same paper. [ABSTRACT FROM AUTHOR]