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This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

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]

In this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

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]

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of 'suggesting' a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

In this paper, we study the non‐Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (Selecta Math. (N.S.) 4 (1998) 279–320). [ABSTRACT FROM AUTHOR]

Given a collection of hypergraphs H=(H1,...,Hm)${{\bf H}}=(H_1, \ldots , H_m)$ with the same vertex set, an m$m$‐edge graph F⊂∪i∈[m]Hi$F\subset \cup _{i\in [m]}H_i$ is a transversal if there is a bijection ϕ:E(F)→[m]$\phi :E(F)\rightarrow [m]$ such that e∈E(Hϕ(e))$e\in E(H_{\phi (e)})$ for each e∈E(F)$e\in E(F)$. How large does the minimum degree of each Hi$H_i$ need to be so that H${{\bf H}}$ necessarily contains a copy of F$F$ that is a transversal? Each Hi$H_i$ in the collection could be the same hypergraph, hence the minimum degree of each Hi$H_i$ needs to be large enough to ensure that F⊆Hi$F\subseteq H_i$. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac‐type results for (powers of) Hamilton cycles. For example, we derive that any collection of rn$rn$ graphs on an n$n$‐vertex set, each with minimum degree at least (r/(r+1)+o(1))n$(r/(r+1)+o(1))n$, contains a transversal copy of the r$r$th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for d ≥ 1 , there exist C 0 , C 1 > 0 , depending on ‖ m ‖ L 1 only, such that C 0 d ≤ supremum of total population ≤ C 1 d. However, when replaced by random diffusion, a conjecture, proposed by Wei-Ming Ni and justified in Bai et al. (Proc. Am. Math. Soc. 144:2161–2170, 2016), indicates that in the one-dimensional case, supremum of total population = 3 ‖ m ‖ L 1. This reflects serious discrepancies between models with local and nonlocal dispersal strategies. Furthermore, we provide an equivalent characterization about the combination of resource distribution and diffusion rate such that the corresponding total population could reach the optimal order d as d goes to infinity. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

We study the existence of nontrivial unbounded surfaces S ⊂ ℝ³ with the property that the constant charge distribution on S is an electrostatic equilibrium, i.e., the resulting electrostatic force is normal to the surface at each point on S . Among bounded regular surfaces S, only the round sphere has this property by a result of Reichel [Arch. Ration. Mech. Anal., 137 (1997), pp. 381--394] (see also Mendez and Reichel [Forum Math., 12 (2000), pp. 223--245]) confirming a conjecture of Gruber. In the present paper, we show the existence of nontrivial exceptional domains Ω ⊂ ℝ³ whose boundaries S=∂Ω enjoy the above property. [ABSTRACT FROM AUTHOR]

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré's works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré's works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré's stated motivations for topology and the Poincaré conjecture. [ABSTRACT FROM AUTHOR]

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. [ABSTRACT FROM AUTHOR]

The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial. [ABSTRACT FROM AUTHOR]

For a simple graph G, the signless Laplacian matrix of G, denoted by Q(G), is defined as D(G) + A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of Q(G). Let T(n, t) be the Turán graph on n vertices and t parts. In De Lima et al. (The clique number and the smallest Q-eigenvalue of graphs. Discrete Math. 2016;339:1744–1752), the authors posed the following conjecture:Conjecture. Let t ≥ 3 and let n be sufficiently large. If G is a Kt+1-free graph of order n and G ≠ T(n, t), then q′(G) < q′(T(n, t)).In this paper, first we disprove the above conjecture and then pose a new conjecture. [ABSTRACT FROM AUTHOR]

Kim and Vu made the following conjecture (Advances in Mathematics, 2004): if d ≫ log n , then the random d-regular graph G (n , d) can asymptotically almost surely be "sandwiched" between G (n , p 1) and G (n , p 2) where p 1 and p 2 are both (1 + o (1)) d / n . They proved this conjecture for log n ≪ d ⩾ n 1 / 3 - o (1) , with a defect in the sandwiching: G (n , d) contains G (n , p 1) perfectly, but is not completely contained in G (n , p 2) . The embedding G (n , p 1) ⊆ G (n , d) was improved by Dudek, Frieze, Ruciński and Šileikis to d = o (n) . In this paper, we prove Kim–Vu's sandwich conjecture, with perfect containment on both sides, for all d where min { d , n - d } ≫ n / log n . The sandwich theorem allows translation of many results from G (n , p) to G (n , d) such as Hamiltonicity, the chromatic number, the diameter, etc. It also allows translation of threshold functions of phase transitions from G (n , p) to bond percolation of G (n , d) . In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability of small subgraph appearances in a random factor of a pseudorandom graph, which is of independent interest. [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]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [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]

The g -extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464–473] determined the 3 -extra edge-connectivity of balanced hypercubes B H n and conjectured that the g -extra edge-connectivity of B H n is λ g ( B H n) = 2 (g + 1) n − 4 g + 4 for 2 ≤ g ≤ 2 n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 1 2 g + 1 and 2 ≤ g ≤ 8 , and disprove their conjecture for n ≥ 3 e g ( B H n) g + 1 and 9 ≤ g ≤ 2 n − 1 , where e g ( B H n) = max { | E ( B H n [ U ]) | | U ⊆ V (B H n) , | U | = g + 1 }. [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 (G) and e(G) denote the domination number and exponential domination number of graph G, respectively. Henning et al., in [Hered-itary equality of domination and exponential domination, Discuss. Math. Graph Theory 38 (2018) 275-285] gave a conjecture: There is a finite set F of graphs such that a graph G satisfies (H) = e(H) for every induced subgraph H of G if and only if G is F-free. In this paper, we study the conjecture for subcubic graphs. We characterize the class F by minimal forbidden induced subgraphs and prove that the conjecture holds for subcubic graphs. [ABSTRACT FROM AUTHOR]

In this paper, we apply our previous estimates in Chen and Cheng [ On the constant scalar curvature Kähler metrics (I): a priori estimates , Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of K-energy in terms of L1 geodesic distance d1 in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the K-energy in (E1, d1) are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized L1 geodesic ray where the K-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs. [ABSTRACT FROM AUTHOR]

A graph G is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum as G. Let k, ti (1 ≤ i ≤ k), and s be natural numbers. A path-friendship graph, Gs,t1...,tk, is a graph of order n = 2s + t1 + ... + tk+1 which consists of s triangles and k paths of lengths t1,t2 ..., tk sharing a common vertex. In this paper, we show that these graphs are DQS and using this result, we respond to a conjecture in [F. Wen, Q. Huang, X. Huang and F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure Appl. Math. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]

The generalized k -connectivity κ k (G) of a graph G , introduced by Hager in 1985, is a natural generalization of the concept of connectivity κ (G) , which is just for k = 2. Rongxia Hao et al. determined the generalized 4-connectivity of the total graphs of the complete equipartition bipartite graph and conjectured that κ 4 (T (K m , m + 1)) = 2 m − 1 and κ 4 (T (K m , n)) = 2 m for n > m + 1 and m ≥ 4 in [Appl. Math. Comput. 422 (2022)]. In this paper, we solved this conjecture. [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]