A connected graph \(G\) is said to be odd path extendable if for any odd path \(P\) of \(G\), the graph \(G – V(P)\) contains a perfect matching. In this paper, we at first time introduce the concept of odd path extendable graphs. Some simple necessary and sufficient conditions for a graph to be odd path extendable are given. In particular, we show that if a graph is odd path extendable, it is hamiltonian.

In this paper, we give one construction for constructing large harmonious graphs from smaller ones. Subsequently, three families of graphs are introduced and some members of them are shown to be or not to be harmonious.

A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that all graphs are set reconstructible if all \(2\)-connected graphs \(G\) with \(diam(G) = 2\) and all \(2\)-connected graphs \(G\) with \(diam(G) = diam(\overline{G}) = 3\) are set reconstructible.

A function \(f: V(G) \to \{-1,0,1\}\) defined on the vertices of a graph \(G\) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every \(v \in V\), \(f(N(v)) \geq 1\), where \(N(v)\) consists of every vertex adjacent to \(v\). The weight of a MTDF is the sum of its function values over all vertices. A MTDF \(f\) is minimal if there does not exist a MTDF \(g: V(G) \to \{-1,0,1\}\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\). The upper minus total domination number, denoted by \(\Gamma^{-}_{t}(G)\), of \(G\) is the maximum weight of a minimal MTDF on \(G\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. The signed total domination number, denoted by \(\gamma^{s}_{t}(G)\), of \(G\) is the minimum weight of a STDF on \(G\). In this paper, we establish an upper bound on \(\Gamma^{-}_{t}(G)\) of the 5-regular graph and characterize the extremal graphs attaining the upper bound. Also, we exhibit an infinite family of cubic graphs in which the difference \(\Gamma^{-}_t(G) – \gamma^{s}_t(G)\) can be made arbitrarily large.

Let \(G\) be a graph with vertex set \(V(G)\). An edge coloring \(C\) of \(G\) is called an edge-cover coloring, if for each color, the edges assigned with it form an edge cover of \(G\). The maximum positive integer \(k\) such that \(G\) has a \(k\)-edge-cover coloring is called the edge cover chromatic index of \(G\) and is denoted by \(\chi’_c(G)\). It is well known that \(\min\{d(v) – \mu(v) : v \in V(G)\} \leq \chi’_c(G) \leq \delta(G)\), where \(\mu(v)\) is the multiplicity of \(v\) and \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’_c(G) = \delta(G)\), \(G\) is called a graph of CI class, otherwise \(G\) is called a graph of CII class. In this paper, we give a new sufficient condition for a nearly bipartite graph to be of CI class.

Though the well-known Vizing’s conjecture is not true for directed graphs in general, we show that it is true when the digraph and its reversal contain an efficient dominating set. In this paper, we investigate the existence of such sets in directed tori and infinite grids. We give a complete characterization of efficient dominating sets in the \(3\)-dimensional case and show the nonexistence of efficient \(d\)-dominating sets in directed tori for any \(d > 1\) and any dimension \(n > 1\).

For every two vertices \(u\) and \(v\) in a graph \(G\), a \(u-v\) geodesic is a shortest path between \(u\) and \(v\). Let \(I(u,v)\) denote the set of all vertices lying on a \(u-v\) geodesic. For a vertex subset \(S\), let \(I_G(S)\) denote the union of all \(I_G(u,v)\) for \(u,v \in S\). The geodetic number \(g(G)\) of a graph \(G\) is the minimum cardinality of a set \(S\) with \(I_G(S) = V(G)\). For a digraph \(D\), there is analogous terminology for the geodetic number \(g(D)\). The geodetic spectrum of a graph \(G\), denoted by \(S(G)\), is the set of geodetic numbers over all orientations of graph \(G\). The lower geodetic number is \(g^-(G) = \min S(G)\) and the upper geodetic number is \(g^+(G) = \max S(G)\). The main purpose of this paper is to investigate lower and upper geodetic numbers of graphs. Our main results in this paper are:

1. For every spanning tree \(T\) of a connected graph \(G\), \(g^-(G) \leq \ell(T)\), where \(\ell(T)\) is the number of leaves of \(T\).
2. The conjecture \(g^+(G) \geq g(G)\) is true for chordal graphs, triangle-free graphs and \(4\)-colorable graphs.

We estimate the essential norm of the weighted composition operator \(uC_{\varphi}\) from the weighted Bergman space \(A^{p}_{\alpha}(\mathbb{B})\) to the weighted space \(H^{\infty}_{\mu}(\mathbb{B})\) on the unit ball \(\mathbb{B}\), when \(p > 1\) and \(\alpha \geq -1\) (for \(\alpha = -1\), \(A^{p}_{\alpha}\) is the Hardy space \(H^{p}(\mathbb{B})\)). We also give a necessary and sufficient condition for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu}(B)\) to be compact, and for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu,0}(\mathbb{B})\) to be bounded or compact, when \(p > 0\), \(\alpha \geq -1\).

Let \(G = (V,E)\) be a graph. A set \(S \subseteq V\) is called a restrained dominating set of \(G\) if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V – S\). The restrained domination number of \(G\), denoted by \(\gamma_r(G)\), is the minimum cardinality of a restrained dominating set of \(G\). In this paper, we establish an upper bound on \(\gamma_r(G)\) for a connected graph \(G\) by the probabilistic method.

Any vertex labeling \(f: V \to \{0,1\}\) of the graph \(G = (V,E)\) induces a partial edge labeling \(f^*: E \to \{0,1\}\) defined by \(f^*(uv) = f(u)\) if and only if \(f(u) = f(v)\). The balance index set of \(G\) is defined as \(\{|f^{*{-1}}(0) – f^{*{-1}}(1)|: |f^{-1}(0) – f^{-1}(1)| \leq 1\}\). In this paper, we first determine the balance index sets of rooted trees of height not exceeding two, thereby completely settling the problem for trees with diameter at most four. Next we show how to extend the technique to rooted trees of any height, which allows us to derive a method for determining the balance index set of any tree.

We show that partial permutation decoding can be used, and give explicit \(s\)-PD-sets in the symmetric group, where \(s\) is less than the full error-correction capability of the code, for some classes of binary codes obtained from the adjacency matrices of the graphs with vertices the \(\binom{n}{3}\) \(3\)-subsets of a set of size \(n\) with adjacency defined by the vertices as \(3\)-sets being adjacent if they have a fixed number of elements in common.

Let \(G\) be a simple connected graph. For a subset \(S\) of \(V(G)\) with \(|S| = 2n + 1\), let \(\alpha_{(2n+1)}(G,S)\) denote the graph obtained from \(G\) by contracting \(S\) to a single vertex. The graph \(\alpha_{(2n+1)}(G, S)\) is also said to be obtained from \(G\) by an \(\alpha_{(2n+1)}\)-contraction. For pairwise disjoint subsets \(S_1, S_2, \ldots, S_{2n}\) of \(V(G)\), let \(\beta_n(G, S_1, S_2, \ldots, S_{2n})\) denote the graph obtained from \(G\) by contracting each \(S_i\) (\(i = 1, 2, \ldots, 2n\)) to a single vertex respectively. The graph \(\beta_{2n}(G, S_1, S_2, \ldots, S_{2n})\) is also said to be obtained from \(G\) by a \(\beta_{2n}\)-contraction. In the present paper, based on \(\alpha_{(2n+1)}\)-contraction and \(\beta_{2}\)-contraction, some new characterizations for \(n\)-extendable bipartite graphs are given.

A graph \(G\) is quasi-claw-free if it satisfies the property: \(d(x, y) = 2 \Rightarrow\) there exists \(u \in N(x) \cap N(y)\) such that \(N[u] \subseteq N[x] \cup N[y]\). In this paper, we prove that the circumference of a \(2\)-connected quasi-claw-free graph \(G\) on \(n\) vertices is at least \(\min\{3\delta + 2, n\}\) or \(G \in \mathcal{F}\), where \(\mathcal{F}\) is a class of nonhamiltonian graphs of connectivity \(2\). Moreover, we prove that if \(n \leq 40\), then \(G\) is hamiltonian or \(G \in \mathcal{F}\).

Let \(K_{n,n}\) denote the complete bipartite graph with \(n\) vertices in each part. In this paper, it is proved that there is no cyclic \(m\)-cycle system of \(K_{n,n}\) for \(m \equiv 2 \pmod{4}\) and \(n \equiv 2 \pmod{4}\). As a consequence, necessary and sufficient conditions are determined for the existence of cyclic \(m\)-cycle systems of \(K_{n,n}\) for all integers \(m \leq 30\).

We examine a design \(\mathcal{D}\) and a binary code \(C\) constructed from a primitive permutation representation of degree \(2025\) of the sporadic simple group \(M^c L\). We prove that \(\text{Aut}(C) = \text{Aut}(\mathcal{D}) = M^c L\) and determine the weight distribution of the code and that of its dual. In Section \(6\) we show that for a word \(w_i\) of weight \(7\), where \(i \in \{848, 896, 912, 972, 1068, 1100, 1232, 1296\}\) the stabilizer \((M^\circ L)_{w_i}\) is a maximal subgroup of \(M^\circ L\). The words of weight \(1024\) split into two orbits \(C_{(1024)_1}\) and \(C_{(1024)_2}\), respectively. For \(w_i \in C_{(1024)_1}\), we prove that \((M^c L)_{w_i}\) is a maximal subgroup of \(M^c L\).

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v, G, \lambda)\)-PD \(((v, G,\lambda)\)-CD), is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(\mathcal{B}\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we have completely determined the packing number and covering number for the graphs with seven points, seven edges and an even cycle.

In this paper, it is shown that there are exactly \(5\) non-isomorphic abstract ovals of order \(9\), all of them projective. The result has been obtained via an exhaustive search, based on the classification of the \(1\)-factorizations of the complete graph with \(10\) vertices.

A graph \(G\) is said to be \(k\)-degenerate if for every induced subgraph \(H\) of \(G\), \(\delta(H) \leq k\). Clearly, planar graphs without \(3\)-cycles are \(3\)-degenerate. Recently, it was proved that planar graphs without \(5\)-cycles or without \(6\)-cycles are also \(3\)-degenerate. And for every \(k = 4\) or \(k \geq 7\), there exist planar graphs of minimum degree \(4\) without \(k\)-cycles. In this paper, it is shown that each \(C_7\)-free plane graph in which any \(3\)-cycle is adjacent to at most one triangle is \(3\)-degenerate. So it is \(4\)-choosable.

This paper investigates the embedding problem for resolvable group divisible designs with block size \(3\). The necessary and sufficient conditions are determined for all \(\lambda \geq 1\).

We provide combinatorial arguments of some relations between classical Stirling numbers of the second kind and two refinements of these numbers gotten by introducing restrictions to the distances among the elements in each block of a finite set partition.

We provide many new edge-magic and vertex-magic total labelings for the cycles \(C_{nk}\), where \(n \geq 3\) and \(k \geq 3\) are both integers and \(n\) is odd. Our techniques are of interest since known labelings for \(C_{k}\) are used in the construction of those for \(C_{nk}\). This provides significant new evidence for a conjecture on the possible magic constants for edge-magic and vertex-magic cycles.

A total dominating set of a graph \(G\) with no isolated vertex is a set \(S\) of vertices of \(G\) such that every vertex is adjacent to a vertex in \(S\). The total domination number of \(G\) is the minimum cardinality of a total dominating set in \(G\). In this paper, we present several upper bounds on the total domination number in terms of the minimum degree, diameter, girth, and order.

We denote by \((p, q)\)-graph \(G\) a graph with \(p\) vertices and \(q\) edges. An edge-magic total (EMT) labeling on a \((p,q)\)-graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow [1,2,\ldots,p+q]\) with the property that, for each edge \(xy\) of \(G\), \(\lambda(x) + \lambda(xy) + \lambda(y) = k\), for a fixed positive integer \(k\). Moreover, \(\lambda\) is a super edge-magic total labeling (SEMT) if it has the property that \(\lambda(V(G)) = \{1, 2,\ldots,p\}\). A \((p,q)\)-graph \(G\) is called EMT (SEMT) if there exists an EMT (SEMT) labeling of \(G\). In this paper, we propose further properties of the SEMT graph. Based on these conditions, we will give new theorems on how to construct new SEMT (bigger) graphs from old (smaller) ones. We also give the SEMT labeling of \(P_n \cup P_{n+m}\) for possible magic constants \(k\) and \(m = 1, 2\),or \(3\).

A Kirkman packing design \(KPD({w, s^*, t^*}, v)\) is a Kirkman packing with maximum possible number of parallel classes, such that each parallel class contains one block of size \(s\), one block of size \(t\) and all other blocks of size \(w\). A \((k, w)\)-threshold scheme is a way of distributing partial information (shadows) to \(w\) participants, so that any \(k\) of them can determine a key easily, but no subset of fewer than \(k\) participants can calculate the key. In this paper, the existence of a \(KPD({3, 4^*, 5^*}, v)\) is established for every \(v \equiv 3 \pmod{6}\) with \(v \geq 51\). As its consequence, some new \((2, w)\)-threshold schemes have been obtained.

In this paper, we mainly define a semidirect product version of the Schützenberger product and also a new two-sided semidirect product construction for arbitrary two monoids. Then, as main results, we present a generating and a relator set for these two products. Additionally, to explain why these products have been defined, we investigate the regularity for the semidirect product version of Schützenberger products and the subgroup separability for this new two-sided semidirect product.

We consider the connected graphs with a unique vertex of maximum degree \(3\). Two subfamilies of such graphs are characterized and ordered completely by their indices. Moreover, a conjecture about the complete ordering of all graphs in this set is proposed.

Let \(G = (V(G), E(G))\) be a simple graph and \(T(G)\) be the set of vertices and edges of \(G\). Let \(C\) be a \(k\)-color set. A (proper) total \(k\)-coloring \(f\) of \(G\) is a function \(f: T(G) \rightarrow C\) such that no adjacent or incident elements of \(T(G)\) receive the same color. For any \(u \in V(G)\), denote \(C(u) = \{f(u)\} \cup \{f(uv) | uv \in E(G)\}\). The total \(k\)-coloring \(f\) of \(G\) is called the adjacent vertex-distinguishing if \(C(u) \neq C(v)\) for any edge \(uv \in E(G)\). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number \(\chi_{at}(G)\) of \(G\). Let \(G\) be a connected graph. If there exists a vertex \(v \in V(G)\) such that \(G – v\) is a tree, then \(G\) is a \(1\)-tree. In this paper, we will determine the adjacent vertex-distinguishing total chromatic number of \(1\)-trees.

In this paper, we extend the study on packing and covering of complete directed graph \(D_t\) with Mendelsohn triples \([6]\). Mainly, the maximum packing of \(D_t-P\) and \(D_t\cup{P}\) with Mendelsohn triples are obtained respectively, where \(P\) is a vertex-disjoint union of directed cycles in \(D_t\).

In the theory of orthogonal arrays, an orthogonal array is called schematic if its rows form an association scheme with respect to Hamming distances. Which orthogonal arrays are schematic orthogonal arrays and how to classify them is an open problem proposed by Hedayat et al. \([12]\). In this paper, we study the Hamming distances of the rows in orthogonal arrays and construct association schemes according to the distances. The paper gives the partial solution of the problem by Hedayat et al. for symmetric and some asymmetric orthogonal arrays of strength two.

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI index of gated amalgam.

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we give formulae to calculate the nullity of \(n\)-vertex bicyclic graphs by means of the maximum matching number.

This note calculates the essential norm of a recently introduced integral-type operator from the Hilbert-Bergman weighted space \(A^2_\alpha(\mathbb{B}), \alpha \geq -1\) to a Bloch-type space on the unit ball \(\mathbb{B} \subset \mathbb{C}^n\).

Let \(G\) be a graph and let \(\sigma_k(G)\) be the minimum degree sum of an independent set of \(k\) vertices. For \(S \subseteq V(G)\) with \(|S| \geq k\), let \(\Delta_k(S)\) denote the maximum value among the degree sums of the subset of \(k\) vertices in \(S\). A cycle \(C\) of a graph \(G\) is said to be a dominating cycle if \(V(G \setminus C)\) is an independent set. In \([2]\), Bondy showed that if \(G\) is a \(2\)-connected graph with \(\sigma_3(G) \geq |V(G)| + 2\), then any longest cycle of \(G\) is a dominating cycle. In this paper, we improve it as follows: if \(G\) is a 2-connected graph with \(\Delta_3(S) \geq |V(G)| + 2\) for every independent set \(S\) of order \(\kappa(G) + 1\), then any longest cycle of \(G\) is a dominating cycle.

Let \(B\) be an \(m \times n\) array in which each symbol appears at most \(k\) times. We show that if \(k \leq \frac{n(n-1)}{8(m+n-2)} + 1\) then \(B\) has a transversal.

Let \(T\) be a partially ordered set whose Hasse diagram is a binary tree and let \(T\) possess a unique maximal element \(1_T\). For a natural number \(n\), we compare the number \(A_T^n\) of those chains of length \(n\) in \(T\) that contain \(1_T\) and the number \(B_T^n\) of those chains that do not contain \(1_T\). We show that if the depth of \(T\) is greater or equal to \(2n + [ n \log n ]\) then \(B_T^n > A_T^n\).

The boundedness and compactness of the weighted composition operator from logarithmic Bloch spaces to a class of weighted-type spaces are studied in this paper.

S.M. Lee proposed the conjecture: for any \(n > 1\) and any permutation \(f\) in \(S(n)\), the permutation graph \(P(P_n, f)\) is graceful. For any integer \(n > 1\), we discuss gracefulness of the permutation graphs \(P(P_n, f)\) when \(f = (123), (n-2, n-1, n), (i, i+1), 1 \leq i \leq n-1, (12)(34)\ldots(2m-1, 2m), 1 \leq m \leq \frac{n}{2}\), and give some general results.

A double-loop network (DLN) \(G(N;r,s)\) is a digraph with the vertex set \(V = \{0,1,\ldots, N-1\}\) and the edge set \(E=\{v \to v+r \pmod{N} \text{ and } v \to v+s \pmod{N} | v \in V\}\). Let \(D(N;r,s)\) be the diameter of \(G(N;r,s)\) and let us define \(D(N) = \min\{D(N;r,s) | 1 \leq r < s < N \text{ and } \gcd(N,r,s) = 1\}\), \(D_1(N) = \min\{D(N;1,s) | 1 < s 0\)). Coppersmith proved that there exists an infinite family of \(N\) for which the minimum diameter \(D(N) \geq \sqrt{3N} + c(\log N)^{\frac{1}{4}}\), where \(c\) is a constant.

In this paper, we consider cycle-partition problems which deal with the case when both vertices and edges are specified and we require that they should belong to different cycles. Minimum degree and degree sum conditions are given, which are best possible.

In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices.

We correct and improve results from a recent paper by G. Ren and U. Kahler, which characterizes the Bloch, the little Bloch and Besov space of harmonic functions on the unit ball \({B} \subset \mathbb{R}^n\).

In a given graph \(G\), a set \(S\) of vertices with an assignment of colors is a defining set of the vertex coloring of \(G\), if there exists a unique extension of the colors of \(S\) to a \(\chi(G)\)-coloring of the vertices of \(G\). A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by \(d(G, \chi)\). Let \(d(n, r, \chi = k)\) be the smallest defining number of all \(r\)-regular \(k\)-chromatic graphs with \(n\) vertices. Mahmoodian \(et.\; al [7]\) proved that, for a given \(k\) and for all \(n \geq 3k\), if \(r \geq 2(k-1)\) then \(d(n, r, \chi = k) = k-1\). In this paper we show that for a given \(k\) and for all \(n < 3k\) and \(r \geq 2(k – 1)\), \(d(n, r, \chi = k) = k-1\).