We present a bijective proof of the hook length formula for rooted trees based on the ideas of the bijective proof of the hook length formula for standard tableaux by Novelli, Pak, and Stoyanovskii \([10]\). In Section \(4\), we present another bijection for the formula.

In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let \(D\) be an \(m\)-colored digraph. We prove that if every chromatic class of \(D\) is quasi-transitive, every cycle is quasi-transitive in the rim and \(D\) does not contain polychromatic triangles, then \(D\) has a kernel by m.d.p. The same result is valid if we preserve the first two conditions before and replace the last one by: there exists \(k \geq 4\) such that every \(\overrightarrow{C}_k\) is quasi-monochromatic and every \(\overrightarrow{C}_{k-1}\) (\(3 \leq l \leq k-1\)) is not polychromatic. Finally, we also show that if every chromatic class of \(D\) is quasi-transitive, every cycle in \(D\) induces a quasi-transitive digraph and \(D\) does not contain polychromatic \(\overrightarrow{C}_3\), then \(D\) has a kernel by m.d.p. Some corollaries are obtained for the existence of kernels by m.d.p. in \(m\)-colored tournaments.

The determination of the zero-capacity of a noisy channel has inspired research on the independence number of the strong product of odd cycles. The independence number for two infinite families of the strong product of three odd cycles is considered in this paper. In particular, we present the independence number of \(C_7 \boxtimes C_9 \boxtimes C_{2k+1}\) and an upper bound on the independence number of \(C_{13} \boxtimes C_3 \boxtimes C_{2k+1}\). The results are partially obtained by a computer search.

The strong product \(G_1 \boxtimes G_2\) of graphs \(G_1\) and \(G_2\) is the graph with \(V(G_1) \times V(G_2)\) as the vertex set, and two distinct vertices \((x_1,x_2)\) and \((y_1,y_2)\) are adjacent whenever for each \(i \in \{1,2\}\) either \(x_i = y_i\) or \(x_iy_i \in E(G_i)\).

An edge irregular total \(k\)-labeling \(\varphi: V \cup E \to \{1,2,\ldots,k\}\) of a graph \(G = (V, E)\) is a labeling of vertices and edges of \(G\) in such a way that for any different edges \(xy\) and \(x’y’\) their weights \(\varphi(x) + \varphi(xy) + \varphi(y)\) and \(\varphi(x’) + \varphi(x’y’) + \varphi(y’)\) are distinct. The total edge irregularity strength, \(\text{tes}(G)\), is defined as the minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling.

We have determined the exact value of the total edge irregularity strength of the strong product of two paths \(P_n\) and \(P_m\).

Given \(m, n\) and \(2 \leq l \leq mn\), we study the problem of separating \(l\) symbols on an \(m \times n\) array such that the minimum \(\ell_1\) distance between any two of the \(l\) symbols is as large as possible. This problem is similar in nature to the well-known Tammes’ problem where one tries to achieve the largest angular separation for a given number of points on a \(2-D\) or higher dimensional sphere. It is also closely related to the well-studied problem of constructing optimal interleaving schemes for correcting error bursts in multi-dimensional digital data where a burst can be an arbitrarily shaped connected region in the array. Moreover, the interest in studying this problem also arises from considerations of minimizing the risk of multiple nearby node failures in a distributed data storage system (or a similar industrial network) in the event of a relatively large scale random disruption. We derive bounds on the maximum possible distance of separation for general \(m,n\) and \(l\), and provide also optimal constructions in several special cases including small and large \(l\) values, small \(m\) (or \(n\)) values, and \(n-1 \geq (l-1)(m-1)\).

In this paper, we apply the concepts of intuitionistic fuzzy sets to coalgebras. We give the definition of intuitionistic fuzzy subcoalgebras and investigate some properties of intuitionistic fuzzy subcoalgebras. Considering the applications of intuitionistic fuzzy subcoalgebras, we discuss their properties under homomorphisms of coalgebras.

In this paper, the joint tree method of graph embeddings, which was introduced by Liu, is generalized to digraph embeddings. The genus distributions of a new type of digraphs in orientable surfaces are determined.

In this paper, the \(m\)-hull sets in the join and composition of two connected graphs are characterized and their \(m\)-hull numbers are shown to be direct consequences of these characterizations.

The generalized de Bruijn digraph denoted by \(G_B(n,m)\) is the digraph \((V, A)\) where \(V = \{0,1,\ldots,m-1\}\) and \((i,j) \in A\) if and only if \(j \equiv ni + \alpha \pmod{m}\) for some \(\alpha \in \{0,1,\ldots,n-1\}\). By replacing each arc of \(G_B(n,m)\) with an undirected edge and eliminating loops and multi-edges, we obtain a generalized undirected de Bruijn graph \(UG_B(n,m)\). In this paper, we prove that the diameter of \(UG_B(n,m)\) is equal to 3 whenever \(n \geq 2\) and \(n^2 + (\frac{\sqrt{5}+1}{2})\leq m \leq 2n^2.\)

The zeroth-order general Randić index of a graph \(G\) is defined as \({}^{0}{}{R}_\alpha = \sum\limits_{v\in V(G)} d(v)^\alpha\)
where \(d(v)\) is the degree of the vertex \(v\) in \(G\) and \(\alpha\) is an arbitrary real number. In the paper, we give sharp lower and upper bounds on the zeroth-order general Randić index of cacti.