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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]

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 = t = d (max T + 1): d | t ⊙ t/d ⊙ T}. We show that espd⊙T (G) = d espT (G) - r, where r, 0 = r = d - 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙T{0}(G) = ⊙d(c(G) - 1), where c(G) is the circular chromatic number of G. This result allows us to formulate several interesting conclusions that include a new formula for the circular chromatic number c(G) = 1 + inf espd⊙{0}(G)/d: d = 1 and a proof that the formula for the T-edge span of powers of cycles, stated as conjecture in [Y. Zhao, W. He and R. Cao, The edge span of T-coloring on graph Cd n, Appl. Math. Lett. 19 (2006) 647-651], is true. [ABSTRACT FROM AUTHOR]

In the classical theory of multiple zeta values (MZV's), Furusho proposed a conjecture asserting that the p-adic MZV's satisfy the same Q -linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho's conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers s = (s 1 , ... , s r) , we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the ∞ -adic MZV and v-adic MZV at s , completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of r = 1 . Furthermore it enables us to apply Yu's sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any ∞ -adic linear relation on MZV's with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV's. [ABSTRACT FROM AUTHOR]

An old conjecture of Erdős and Graham states that only finitely many integer squares could be obtained from product of disjoint blocks of consecutive integers of length greater than or equal to four. It is known by counterexamples that the conjecture is false for product of disjoint blocks of four and five consecutive integers. In this paper, we present new algorithms generating new polynomial parametrizations that extend the polynomial parametrization given by Bennett and Luijk (Indag Math (N.S.) 23(1–2):123–127, 2012). Moreover, we produce the first examples of integer squares obtained from product of disjoint blocks of consecutive integers such that each block has length six or seven. [ABSTRACT FROM AUTHOR]

A simple graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G — v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then E(G) = 2n — 3. A non-increasing sequence π = (d1,...,dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. [Acta Math. Sin. Engl. Ser., 25, 795–802 (2009)] proved that if k ≥ 2, n ≥ and π = (d1,...,dn) is a graphic sequence with , then p has a realization containing every 1-tree (the usual tree) on k vertices. Moreover, the lower bound (k — 2)n is the best possible. This is a variation of a conjecture due to Erdos and Sós. In this paper, we investigate an analogue problem for 2-trees and prove that if k ≥ 3 is an integer with k = i (mod3), π = (d1,...,dn is a graphic sequence with , then π has a realization containing every 2-tree on k vertices. Moreover, the lower bound max is the best possible. This result implies a conjecture due to [Discrete Math. Theor. Comput. Sci., 17(3), 315–326 (2016)]. [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]

Let k be a positive integer and let D be a digraph. A path partition P of D is a set of vertex-disjoint paths which covers V(D). Its k-norm is defined as ∑ P ∈ P min { | V (P) | , k } . A path partition is k-optimal if its k-norm is minimum among all path partitions of D. A partialk-coloring is a collection of k disjoint stable sets. A partial k-coloring C is orthogonal to a path partition P if each path P ∈ P meets min { | V (P) | , k } distinct sets of C . Berge (Eur J Comb 3(2):97–101, 1982) conjectured that every k-optimal path partition of D has a partial k-coloring orthogonal to it. A (path) k-pack of D is a collection of at most k vertex-disjoint paths in D. Its weight is the number of vertices it covers. A k-pack is optimal if its weight is maximum among all k-packs of D. A coloring of D is a partition of V(D) into stable sets. A k-pack P is orthogonal to a coloring C if each set C ∈ C meets min { | C | , k } paths of P . Aharoni et al. (Pac J Math 2(118):249–259, 1985) conjectured that every optimal k-pack of D has a coloring orthogonal to it. A digraph D is semicomplete if every pair of distinct vertices of D are adjacent. A digraph D is locally in-semicomplete if, for every vertex v ∈ V (D) , the in-neighborhood of v induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge's and Aharoni–Hartman–Hoffman's Conjectures for locally in/out-semicomplete digraphs. [ABSTRACT FROM AUTHOR]

The leaf distance of a tree is the maximum d such that the distance between any pair of leaves in the tree is at least d . Kaneko provided sufficient conditions to force the existence of a spanning tree with leaf distance at least d = 3 and conjectured that similar conditions suffice for larger d . The case when d = 4 was later proved by Kaneko, Kano, and Suzuki. In this paper, we show that when d ≥ 4 , a stronger form of this conjecture holds for graphs with independence number at most five. As an immediate corollary, we obtain that when d ≥ n ∕ 3 , this stronger version holds for all n -vertex graphs, consequently proving Kaneko’s conjecture for d ≥ n ∕ 3 . [ABSTRACT FROM AUTHOR]