For a graph $G$ and positive integers $a_1,\dots ,a_r,$ if every r-coloring of vertics V(G) must result in a monochromatic $a_1$-clique of color $i$ for some $i\in \{1,\dots ,r\},$ then we write $G\to (a_1,..a_r{)}^v$.$F_v(K_{a}1,\dots ,K_{a}r;H)$ is the smallest integer $n$ such that there is an H-free graph $G$ of order $n$, and $G\to (a_1,\dots ,a_r{)}^v$. In this paper we study upper and lower bounds for some generalized vertex Folkman numbers of from $F_v(K_{a1},\dots ,K_{ar};K_4–e)$, where $r\in 2,3$ and $a_1\in 2,3$ for 10 and $F_v(K_2,K_3;K_4–e)=19$ by computing, and prove $F_v(K_3,K_3;K_4–e)\ge F_v(K_2,K_2,K_3;K_4–e)\ge 25$

We study the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of a Vertex Cover with bounded size” (U-VC) and “Uniqueness of an Optimal Vertex Cover” (U-OVC), and for any fixed integer $r\ge 1,$ “Uniqueness of an $r$-Dominating Code with bounded size” $(U-D{C}_r)$ and “Uniqueness of an Optimal $r$-Dominating Code” $(U-OD{C}_r$. In particular, we give a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OVC, and from U-SAT to U-ODC, We prove that U-VC and $U-D{C}_r$ have complexity equivalent to that of U-SAT (up to polynomials); consequently, these problems are all $NP$-hard, and U-VC and $U-D{C}_r$ belong to the class $DP$.

Let $D$ be a finite and simple digraph with vertex set $V(D)$. A signed total Roman dominating function on the digraph $D$ is a function $f:V(D)⟶-1,1,2$ $\sum _{u\in N-(v)}f(u)\ge 1$ for every $v\in V(D)$, where $N^{-}(v)$ consists of all inner neighbors of $v$ for dominating function on $D$ with the property that $\sum _d^{i=1}{f}_i(v)\le 1$ for each $v\in V(D)$ is called a signed total roman dominating family (of functions) on $D$. The maximum number of functions in a signed total roman dominating family on $D$is the signed total Roman domatic number of $D$. denoted by $d_{stR}(D)$. In addition, we determine the signed total Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the signed total Roman domatic of graphs.

Let $G=(V,E)$ be a graph. The transitivity of a graph $G$, denoted $Tr(G)$, equals the maximum order $k$ of a partition $\pi =\{V_1,V_2,\dots ,V_k\}$ of $V$ such that for r=every $i,j,1\le i<j\le k,V_i$ dominates $V_j$. We consider the transitivity in many special classes of graphs, including cactus graphs, coronas, Cartesian products, and joins. We also consider the effects of vertex or edge deletion and edge addition on the transivity of a graph.

We dedicate this paper to the memory of professor Bohdan Zelinka for his pioneering work on domative of graphs.

The n-dimensional enhanced hypercube $Q_{n,k}(1\le k\le n-1)$ is one of the most attractive interconnection networks for parallel and distributed computing system. Let $H$ be a certain particular connected subgraph of graph $G$. The $H$-structure-connectivity of $G$, denoted by $\kappa (G;H),$ is the cardinality of minimal set of subgraphs $F=\{H_1,H_2,\dots ,H_m\}$ in $G$ such that every $H_i\in F$ is isomprphic to $H$ and $G-F$ is disconnected. The $H$-substructure-connectivity of $G$, denoted by ${}_k^3(G;H)$, is the cardinality of minimal set of subgraphs $F=H_1,H_2,\dots ,H_m$ in $G$ such that every $H_i\in F$ is isomorphic to a connected subgraph $H$ , and $G-F$ is disconnected. Using the structural properties of $Q_{n,k}$ the $H$-structure-connectivity $\kappa (Q_{n,k};H)$ were determine for $H\in \{K_1,K_{1,1},K_{1,2},K_{1,3}\}$.

In this paper, we characterize the set of spanning trees of $G_{n,r}^1$ (a simple connected graph consisting of $n$ edges, containing exactly one 1-edge-connected chain of $r$ cycles ${\mathbb{C}}_r^1$ and $G_{n,r}^1{\mathbb{C}}_r^1$ is a forest). We compute the Hilbert series of the face ring $k[{\mathrm{\Delta }}_s(G_{n,r}^1)]$ for the spanning simplicial complex ${\mathrm{\Delta }}_s(G_{n,r}^1)$. Also, we characterize associated primes of the facet ideal $I_{\mathcal{F}}({\mathrm{\Delta }}_s(G_{n,r}^1)$. Furthermore, we prove that the face ring $k[{\mathrm{\Delta }}_s(G_{n,r}^1)]$ is Cohen-Macaulay.

We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset $P$. Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points and inducing the poset $P$ on the non-minimal points. It turns out that these counting functions are closely related to a collection of downset numbers of certain subposets. Since any function ar that kind is an exponential sum (with the number of minimal points as exponents), we call it the exponential function of the poset. Some linear equalities, divisibility relations, upper and lower bounds. A list of all such exponential functions for posets with up to five points concludes the paper.

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The first Zagreb index of a graph is defined as the sum of squares of the degrees of the vertices of the graph. The second Zagreb index of a graph is defined as the sum of products of the degrees of a pairs of the adjacent vertices of the graph. In this paper, we establish some sufficient conditions for a nearly balanced bipartite graph with large minimum degree to be traceable in terms of the energy, the first Zagreb index and the second Zagreb index of the quasi-complement of the graph, respectively.

In $PG(3,P^{2^h}),$ ovoids and cones projecting an oval from a point are characterized as three character sets with respect to lines and planes, respectively.

There are 19 connected cubic graphs of order 10. If $G$ is one of a specific set 3 the 19 graphs. we find necessary and sufficient conditions for the existence of $G$-decompositions of $K_v$.

For a positive integer $k,$ let $[k]=1,2,\dots ,k$, let $P([k])$ denote the power set of the set $[k]$ and let $P\ast ([k])=P([k])–\mathrm{\varnothing }$. For each integer $t$ with $1\le t<k$, let $P_t([k])$ denote the set of $t$-element subsets of $P([k])$. For an edge coloring $c:E(G)\to P_t([k])$ of a graph $G$, where adjacent edges may be colored the same, $c^{\prime }:V(G)\to P\ast ([k])$ is the vertex coloring in which $c^{\prime }(v)$ is the union of the color sets of the edges incident with $v$. If $c^{\prime }$ is a proper vertex coloring of $G$, then $c$ is a majestic $t$-tone k-coloring of $G$ For a fixed positive integer $t$, the minimum positive integer $k$ for which a graph $G$ has a majestic t-tone k-coloring is the majestic t-tone index $ma{j}_t(G)$ of $G$. It is known that if $G$ is a connected bipartite graph or order at least 3, then $ma{j}_t(G)=t+1$ or $ma{j}_t(G)=t+2$ for each positive integer t. It is shown that (i) if $G$ is a 2-connected bipartite graph of arbitrarily large order $n$ whose longest cycles have length $l$ where where $n-5\le l\le n$ and $t\ge 2$ is an integer, then $ma{j}_t(G)=t+1$ and (ii) there is a 2-connected bipartite graph F of arbitrarily large order n whose longest cycles have length n-6 and $ma{j}_2(F)=4$. Furthermore, it is shown for integers $k,t\ge 2$ that there exists a k-connected bipartite graph $G$ such that $ma{j}_t(G)=t+2$. Other results and open questions are also presented.

The 3-path $P_3(G)$ of a connected graph $G$ of order 3 or more has the set of all 3-path (path of order 3) of $G$ as its vertex of $P_3(G)$ are adjacent if they have a 2-path in common. A Hamiltonian walk in a nontrivial connected graph $G$ is a closed walk of minimum length that contains every vertex of $G$. With the aid of spanning trees and Hamiltonian walks in graphs, we provide sufficient conditions for the 3-path graph of a connected graph to be Hamiltonian.

Let $R$ be a commutative ring with identity. For any integer $K>1,$ an element is a $k$ zero divisor if there are $K$ distinct elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let $Z(R,K)$ denote the set of the $K$ zero divisors of $R$. A ring with no $K$-zero divisors is called a $K$-domain. In this paper we define the hyper-graphic constant $HG(R)$ and study some basic properties of $K$-domains. Our main results is theorem 5.1 which is as fellow:

Let $R$ be a commutative ring such that the total ring of fraction $T(R)$ is dimensional. If $R$ is a $K$-domain for $k\ge 2,$ then $R$ has finitely many minimal prime ideals.

Using the results and lemma 5.4, we improve a finiteness theorem proved in [11] to a more robust theorem 5.5 which says:

Suppose $R$ is not a $k$-domain and has more then $k$-minimal prime ideals.

Further, suppose that $T(R)$ is a zero dimensional ring. Then $Z(R,K)$ is finite if and only if $R$ is finite.

We end this paper with a proof of an algorithm describing the maximal $k$-zero divisor hypergraphs on ${\mathbb{Z}}_n$.

The subject matter for this paper is GDDs with three groups of sizes $n_1,n(n\ge n_1)$ and $n+1$, for $n_1=1or2$ and block size four. A block having Configuration $(1,1,2)$ means that the block contains 1 point from two different groups and 2 points from the remaining group. a block having Configuration $(2,2)$ means that the block has exactly two points from two of the three groups. First, we prove that a GDD$(n_1,n,n+1,4;{\lambda }_1,{\lambda }_2)$ for $n_1=1or2$ does not exist if we require that exactly halh of the blocks have the Configuration $(1,1,2)$ and the other half of the blocks have the configuration $(2,2)$ except possibly for n=7 when $n_1=2$. Then we provide necessary conditions for the existence of a GDD$(n_1,n,n+1,4;{\lambda }_1,{\lambda }_2)$ for $n_1=1and2$, and prove that these conditions are sufficient for several families of GDDs. For $n_1=2$, these usual necessary conditions are not sufficient in general as we provide a glimpse of challenges which exist even for the case of $n_1=2$. A general results that a GDD$(n_1,n_2,n_3,4;{\lambda }_1,{\lambda }_2)$ exists if $n_1+n_2+n_3=0,4$ $(mod12)$ is also given.

Recently GDDs with two groups and block size four were studied in a paper where the authors constructed two families out of four possible cases with an equal number of even, odd, and group blocks. In this paper, we prove partial existence of one of the two remaining families, namely $GDD(11t+1,2,4;11t+1,7t)$, with 7 $\nmid$(11t+ 1). In addition, we show a useful construction of $GDD(6t+4,2,4;2,3)$.

A cancellable number (CN) is a fraction in which a decimal digit can be removed (“‘canceled”) in the numerator and denominator without changing the value of the number; examples include $\frac{64}{16}$ where the 6 can be canceled and $\frac{98}{49}$ where the 9 can be canceled. We show that the slope of the line of a cancellable number need not be negative.

A complete bipartite graph with the number of two partitions s and t is denoted by $Ks,t$. For a positive integer s and two bipartite graphs G and H, the s-bipartite Ramsey number $B{R}_s(G,H)$ of G and H is the smallest integer t such that every 2-coloring of the edges of $K_{s,t}$ contains the a copy of G with the first color or a copy of H with the second color. In this paper, by using an integer linear program and the solver Gurobi Optimizer 8.0, we determine all the exact values of $B{R}_s(K_{2,3},K_{3,3})$ for all possible $s$. More precisely, we show that $B{R}_s(K_{2,3},K_{3,3})=13$ for $s$ $\in$ {8,9}, $B{R}_s(K_{2,3},K_{3,3})=12$ for $s\in \{10,11\}$, $B{R}_s(K_{2,3},K_{3,3})=10$ for $s=12$, $B{R}_s(K_{2,3},K_{3,3})=8$ for $s\in \{13,14\}$, $B{R}_s(K_{2,3},K_{3,3})=6$ for $s\in \{15,16,\dots ,20\}$, and $B{R}_s(K_{2,3},K_{3,3})=4$ for s ≥ 21. This extends the results presented in [Zhenming Bi, Drake Olejniczak and Ping Zhang, “The s-Bipartite Ramsey Numbers of Graphs $K_{2,3}$ and $K_{3,3}$” , Journal of Combinatorial Mathematics and Combinatorial Computing 106, (2018) 257-272].

In this work we construct many new examples of maximal partial line spreads in $PG(3,q),q$ even. We do this by giving a suitable representation of $PG(3,q)$ in the non-singular quadric $Q(4,q)$ of $PG(4,q)$. We prove the existence of maximal partial line spreads of sizes $q^2-q+1–\overline{t}\overline{z}$, for every pair $(\overline{t},\overline{z})\in P_1\cup P_2$, where $P_1$ and $P_2$ are the pair sets $P_1=\{(t,z)\in \mathbb{Z}\times \mathbb{Z}:\frac{q}{2}-2\le t\le q-3,0\le z\le \frac{q}{2}-2\}$ and $P_2=\{(t,z)\in \mathbb{Z}\times \mathbb{Z}:0\le t\le \frac{q}{2}-3,0\le z\le q-1\}$, for $q\ge 8$.

Low-Density Parity-Check (LDPC) codes have low linear decoding complexity, which is a kind of good codes with excellent performance. Therefore, LDPC codes have great research value. This article is based on vector space over finite field as a theoretical tool by the inclusive relation of vector subspaces to construct protograph, and then constructs the LDPC codes with larger girth based on protograph by the modified progressive edge growth(M-PEG) algorithm, and utilize the related knowledge, such as Anzahl theorem in vector space, determines the code length, code rate and code word number of the LDPC codes. Moreover, the LDPC codes constructed are compared with the existing codes, and the constructed codes are better than some existing ones.

Let G be a graph and a1,…, as be positive integers. The expression $G\underrightarrow{v}(a_1,\dots ,a_s)$ means that for every coloring of the vertices of G in s colors there exists $i\in 1,\dots ,s$ such that there is a monochromatic $a_i$-clique of color i. The vertex Folkman numbers $F_v(a_1,\dots ,a_s;q)$ are defined by the equality:
$$F_v(a_1,\dots ,a_s;q)=min\{|V(G)|:G\underrightarrow{v}(a_1,\dots ,a_s)and{K}_q⊈G\}$$.
Let $m=\sum _{i=1}^s(a_i–1)+1$. It is easy to see that $F_v(a_1,\dots ,a_s;q)=m$ if $q\ge m+1$. In [11] it is proved that $F_v(a_1,\dots ,a_s;m)=m+max{a}_1,\dots ,a_s$. We know all the numbers $F_v(a_1,\dots ,a_s;m–1)$ when $max{a}_1,..,a_s\le 5$ and none of these numbers is known if $max{a}_1,\dots ,a_s\ge 6$. In this paper we present computer algorithms, with the help of which we compute all Folkman numbers $F_v(a_1,..,a_s;m–1)$ when $max{a}_1,..,a_s=6$. We also prove that $F_v(2,2,7;8)=20$ and obtain new bounds on the numbers $F_v(a_1,\dots ,a_s;m–1)$ when $max(a_1,\dots ,a_s)=7$.

Let G be an edge-colored connected graph. For vertices u and v of G, a shortest u – v path P in G is a u – u geodesic and P is a proper u – u geodesic if no two adjacent edges in P are colored the same. An edge coloring of a connected graph G is called a proper k-geodesic coloring of G for some positive integer k if for every two nonadjacent vertices u and v of G, there exist at least k internally disjoint proper u – u geodesics in G. The minimum number of the colors required in a proper k-geodesic coloring of G is the strong proper k-connectivity $sp{c}_k(G)$ of G. Sharp lower bounds are established for the strong proper k-connectivity of complete bipartite graphs $K_{r,s}$ for all integers k, r, s with 2 ≤ k ≤ r ≤ s and it is shown that the strong proper 2-connectivity of $K_{r,s}$ is $sp{c}_2(K_{r,s})=\lceil {}^{r-1}\sqrt{s}\rceil$ for 2 ≤ r ≤ s.

Suppose $G=(V,E)$ is a simple graph and $f:(V\cup E)\to 1,2,\dots ,k$ is a proper total k-coloring of G. Let $C(u)=f(u)\cup f(uv):uv\in E(G)$ for each vertex u of G. The coloring f is said to be an adjacent vertex-distinguishing total coloring of G if $C(u)\ne C(v)$ for every $uv\in E(G)$. The minimum k for which such a chromatic number of G, and is denoted by $X_{a}t(G)$. This paper considers three types of cubic graphs: a specific family of cubic hasmiltonian graphs, snares and Generalized Petersen graphs. We prove that these cubic graphs have the same adjacent vertex-distinguishing total chromatic number 5. This is a step towards a problem that whether the bound $X_{a}t(G)\le 6$ is sharp for a graph G with maximum degree three.

An Italian dominating function (IDF) on a graph G = (V,E) is a function f: V → {0,1,2} satisfying the property that for every vertex $v\in V$, with f(v) = 0, $\sum _{u\in N_{(v)}}f(u)\ge 2$. The weight of an IDF f is the value $w(f)=f(V)=\sum _{u\in V}f(u)$. The minimum weight of an IDF on a graph G is called the Italian domination number of G, denoted by ${\gamma }_I(G)$. For a graph G = (V,E), a double Roman dominating function (or just DRDF) is a function f: V → {0, 1, 2,3} having the property that if $f(v)=O$ for a vertex $u$, then $u$ has at least two neighbors assigned 2 under for one neighbor assigned 3 under f, and if $f(v)=1$, then u has at least one neighbor with f(w) ≥ 2. The weight of a DRDF f is the sum $f(V)=\sum _{u\in V}f(v)$, and the minimum weight of a DRDF on G is the double Roman domination number oi G, denoted by ${\gamma }_{dR}(G)$. In this paper we show that ${\gamma }_{d}R(G)/2\le {\gamma }_I(G)\le 2{\gamma }_{d}R(G)/3$, and characterize all trees T with ${\gamma }_I(T)=2{\gamma }_{d}R(T)/3$.

This paper mainly presents a construction of LDPC codes based on symplectic spaces. By two subspaces of type (m, r) to produce a subspace of type (m + 1,r) or (m + 1,r + 1) in ${\mathbb{F}}^{2v}$ , we use all subspaces of type (m,r) to mark rows and all subspaces of type (m + 1, r) and (m + 1, r + 1) to mark columns of check matrix H. A construction of LDPC codes has been given based on symplectic spaces. As a special case, we use all subspaces of type (1,0) to mark rows and all subspaces of type (2,0) and (2,1) to mark columns of check matrix $H_1$, in ${\mathbb{F}}_q^4$, the cycles of length 6 of $H_1$, is further discussed.

We introduce and study a subring SC of $\mathbb{Z}S{L}_2({\mathbb{F}}_q)$ obtained by summing elements of $S{L}_2({\mathbb{F}}_q)$ according to their support. The ring SC can be used for the construction of several association schemes.

Randic index and geometric-arithmetic index are two important chemical indices. In this paper, we give the generalized Nordhaus-Gaddum-type inequalities for the two kinds of chemical indices.

Graceful graphs were first studied by Rosa [17]. A graceful labeling $f$ of a graph G is a one-to-one map from the set of vertices of $G$ to the set {0.1,., |E(G)|}. where for edges $xy$, the induced edge labels |f(x) – f (y)| form the set {1,2,., |E(G)|, with no label repeated. In this paper, we investigate the set of labels taken by the central vertex of the star in the graph $K_{1.m-1}\oplus C_n$, for each graceful labeling. We also study gracefulness of certain unicyclic graphs where paths $P_3,P_2$ are pendant at vertices of the cycle. For these unicyclic graphs, the deletion of any edge of the cycle does not result in a caterpillar.

An edge-magic total labeling of a graph $G=(V,E)$ is an assignment of integers $1,2,\dots ,|V|+|E|$ to the vertices and edges of the graph so that the sum of the labels of any edge $uv$ and the labels on vertices $u$ and $v$ is constant. It is known that the class of complete graphs on $n$ vertices, $K_n$, are not edge magic for any n ≥ 7. The edge magic number $M_E(K_n)$ is defined to be the minimum number t of isolated vertices such that $K_n\cup t{K}_1$, is edge magic. In this paper we show that, for n ≥ 10, $M_E(K_n)\le f_{n+1}+57–\frac{n^2+n}{2}where\text{(}{f}_i$ is the $i^{th}$ Fibonacci number. With the aid of a computer, we also show that $M_E(K_7)=4,M_E(K_8)=10$, and $M_E(K_9)=19$, answering several questions posed by W. D. Wallis.

A cograph is a simple graph that does not contain an induced path on 4 vertices. A graph G is $k_{-e}colorable$ if the vertices of G can be colored in k colors such that, for each color, the subgraph induced by the vertices assigned the color is a cograph. A graph that is $k_{-e}colorable$ and is not $(k-1{)}_{-e}colorable$, but becomes $(k-1{)}_{-e}colorable$ whenever a vertex is removed, is called $k_{-e}critical$ graph. Two general constructions are provided that produce critical graphs from color critical graphs and hypergraphs. A characterization is also given for when a general composition of graphs (path-joins) is critical. The characterization is used to provide an upper bound for the fewest number of vertices of a $k_{-e}critical$ graph.

We survey Dudeney’s round table problem which asks for a set of Hamilton cycles in the complete graph that uniformly covers the 2-paths of the graph. The problem was proposed about one hundred years ago but it is still unsettled. We mention the history of the problem, known results, gener-alizations, related designs, and some open problems.

Constructions are given for non-cubic, edge-critical Hamilton laceable bigraphs with 3m edges on 2m vertices for all m ≥ 4. The significance of this result is that it shows the conjectured hard upper bound of 3m edges for edge-critical bigraphs on 2m vertices is populated by both cubic and non-cubic cases for all m. This is unlike the situation for the hard 3m-edge lower bound for edge-stable bigraphs where the bound is populated exclusively by cubics.

In this paper, we identify LOW and OLW graphs, find the minimum $\lambda$ for decomposition of $\lambda k_n$, into these graphs, and show that for all viable values of $\lambda$, the necessary conditions are sufficient for LOW- and OLW-decompositions using cyclic decompositions from base graphs.

A maximal independent set is an independent set that is not a proper subset of any other independent set. A twinkle graph W is a connected unicyclic graph with cycle C such that W – x is disconnected for any $x\in V(C)$. In this paper, we determine the largest number of maximal independent sets and characterize those extremal graphs achieving these values among all twinkle graphs. Using the results on twinkle graphs, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs with at most one cycle.

A branch vertex of a tree is a vertex of degree at least three. Matsuda, et. al. [7] conjectured that, if $n$ and $k$ are non-negative integers and $G$ is a connected claw-free graph of order $n$, there is either an independent set on $2k+3$ vertices whose degrees add up to at most $n–3$, or a spanning tree with at most $k$ branch vertices. The authors of the conjecture proved it for $k=1$; we prove it for $k=2$.

Orbit code is a class of constant dimension code which is defined as orbit of a subgroup of the general linear group $G{L}_n(\mathbb{F}q)$, acting on the set of all the subspaces of vector space ${\mathbb{F}}_q^n$. In this paper, the construction of orbit codes based on the singular general linear group $GLn+l,n({\mathbb{F}}_q)$ acting on the set of all the subspaces of type $(m,k)$ in singular linear spaces ${\mathbb{F}}_q^{n+l}$ is given. We give a characterization of orbit code constructed in singular linear space ${\mathbb{F}}_q^{n+l}$, and then give some basic properties of the constructed orbit codes. Finally, two examples about our orbit codes for understanding these properties explicitly are presented.

We use heuristic algorithms to find terraces for small groups. We show that Bailey’s Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell’s Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups $A_6,S_6,PSL(2,q_1)$ and $PGL(2,q_2)$ where $q_1$ and $q_2$ are prime powers with $3\le q_1\le 11$ and $3\le q_2\le 8$. A sequencing for a group of a given order implies the existence of a complete Latin square at that order. We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. This gives 31 new orders at which complete Latin squares are now known to exist, the smallest of which is 63.

In addition, we consider terraces with some special properties, including constructing a directed $T_2$-terrace for the non-abelian group of order 21 and hence a Roman-2 square of order 21 (the first known such square of odd order). Finally, we report the total number of terraces and directed terraces for groups of order at most 15.

An adjacent vertex distinguishing total coloring of a graph $G$ is a proper total coloring of $G$ such that no pair of adjacent vertices are incident to the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of $G$ is denoted by $\chi {”}_a(G)$. In this paper, we prove that if $G$ is an outer 1-planar graph with at least two vertices, then $\chi {”}_a(G)\le max\{\mathrm{\Delta }+2,8\}$. Moreover, we also prove that when $\mathrm{\Delta }\ge 7$, $\chi {”}_a(G)=\mathrm{\Delta }+2$ if and only if $G$ contains two adjacent vertices of maximum degree.

In this paper, we study five methods to construct $\alpha$-trees by using vertex amalgamations of smaller $\alpha$-trees. We also study graceful and $\alpha$-labelings for graphs that are the union of $t$ copies of an $\alpha$-graph $G$ of order $m$ and size $n$ with a graph $H$ of size $t$. If $n>m$, we prove that the disjoint union of $H$ and $t$ copies of $G$ is graceful when $H$ is graceful, and that this union is an $\alpha$-graph when $H$ is any linear forest of size $t–1$. If $n=m$, we prove that this union is an $\alpha$-graph when $H$ is the path $P_{t-1}$.

For a bipartite graph $G$ and a positive integer $s$, the $s$-bipartite Ramsey number $B{R}_s(G)$ of $G$ is the minimum integer $t$ with $t\ge s$ for which every red-blue coloring of $K_{s,t}$ results in a monochromatic $G$. A formula is established for $B{R}_s(K_{r,r})$ when $s\in \{2r–1,2r\}$ when $r\ge 2$. The $s$-bipartite Ramsey numbers are studied for $K_{3,3}$ and various values of $s$. In particular, it is shown that $B{R}_5(K_{3,3})=41$ when $s\in \{5,6\}$, $B{R}_s(K_{3,3})=29$ when $s\in \{7,8\}$, and $17\le B{R}_{10}(K_{3,3})\le 23$.

Research collaboration is a central mechanism that combines distributed knowledge and expertise into common new original ideas. Considering the lists of publications of László Lovász from the Hungarian bibliographic database MTMT, we illustrate and analyze the collaboration network determined by all co-authors of Lovász, considering only their joint works with Lovász.

In the second part, we construct and analyze the co-authorship network determined by the collaborating authors of all scientific documents that have referred to Lovász according to the Web of Science citation service. We study the scientific influence of László Lovász as seen through this collaboration network. Here, we provide some statistical features of these publications, as well as the characteristics of the co-authorship network using standard social network analysis techniques.

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. If $k\ge 2$ is an integer, then the signed edge $k$-independence function of $G$ is a function $f:E\to \{-1,1\}$ such that $\sum _{e^{\prime }\in N[e]}f(e^{\prime })\le k–1$ for each $e\in E$. The weight of a signed edge $k$-independence function $f$ is $\omega (f)=\sum _{e\in E}f(e).$ The signed edge $k$-independence number ${\alpha }_k^s(G)$ of $G$ is the maximum weight of a signed edge $k$-independence function of $G$. In this paper, we initiate the study of the signed edge $k$-independence number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.

Let $S=S_1{S}_2{S}_3\dots S_n$ be a finite string which can be written in the form $X_1^{k_1}{X}_2^{k_2}\dots X_r^{k_r}$, where $X_i^{k_i}$ is the $k_i$ copies of a non-empty string $X_i$ and each $k_i$ is a non-negative integer. Then, the curling number of the string $S$, denoted by $\text{cn}(S)$, is defined to be $\text{cn}(S)=max\{k_i:1\le i\le r\}$. Analogous to this concept, the degree sequence of the graph $G$ can be written as a string $X_1^{k_1}\circ X_2^{k_2}\circ X_3^{k_3}\circ \cdots \circ X_r^{k_r}$. The compound curling number of $G$, denoted ${\text{cn}}^c(G)$, is defined to be ${\text{cn}}^c(G)=\prod _{i=1}^r{k}_i.$ In this paper, the curling number and compound curling number of the powers of the Mycielskian of certain graphs are discussed.

The symmetric inverse monoid, $\text{SIM}(n)$, is the set of all partial one-to-one mappings from the set $\{1,2,\dots ,n\}$ to itself under the operation of composition. Earlier research on the symmetric inverse monoid delineated the process for determining whether an element of $\text{SIM}(n)$ has a $k$th root. The problem of enumerating $k$th roots of a given element of $\text{SIM}(n)$ has since been posed, which is solved in this work. In order to find the number of $k$th roots of an element, all that is needed is to know the cycle and path structure of the element. Conveniently, the cycle and cycle-free components may be considered separately in calculating the number of $k$th roots. Since the enumeration problem has been completed for the symmetric group, this paper only focuses on the cycle-free elements of $\text{SIM}(n)$. The formulae derived for cycle-free elements of $\text{SIM}(n)$ here utilize integer partitions, similar to their use in the expressions given for the number of $k$th roots of permutations.

Motivated by finding a way to connect the Roman domination number and 2-domination number, which are in general not comparable, we consider a parameter called the Italian domination number (also known as the Roman $(2)$-domination number). This parameter is bounded above by each of the other two. Bounds on the Italian domination number in terms of the order of the graph are shown. The value of the Italian domination number is studied for several classes of graphs. We also compare the Italian domination number with the 2-domination number.

The 3-sphere regular cellulation conjecture claims that every 2-connected cyclic graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere. The conjecture is obviously true for planar graphs. 2-connectivity is a necessary condition for a graph to satisfy such a property. Therefore, the question whether a graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere would be equivalent to the 2-connectivity test if the conjecture were proved to be true. On the contrary, it is not even clear whether such a decision problem is computationally tractable.

We introduced a new class of graphs called weakly-split and proved the conjecture for such a class. Hamiltonian, split, complete $k$-partite, and matrogenic cyclic graphs are weakly split. In this paper, we introduce another class of graphs for which the conjecture is true. Such a class is a superclass of planar graphs and weakly-split graphs.

The maximum number of internal disjoint paths between any two distinct nodes of faulty enhanced hypercube $Q_{n,k}(1\le k\le n-1)$ are considered in a more flexible approach. Using the structural properties of $Q_{n,k}(1\le k\le n-1)$, $min(d_{Q_{n,k}-V}(x),d_{Q_{n,k}-V}(y))$ disjoint paths connecting two distinct vertices $x$ and $y$ in an $n$-dimensional faulty enhanced hypercube $Q_{n,k}-V(n\ge 8,k\ne n-2,n-1)$ are conformed when $|V^{\prime }|$ is at most $n-1$. Meanwhile, it is proved that there exists $min(d_{Q_{n,k}-V}(x),d_{Q_{n,k}-V}(y))$ internal disjoint paths between $x$ and $y$ in $Q_{n,k}-V(n\ge 8,k\ne n-2,n-1)$, under the constraints that (1) The number of faulty vertices is no more than $2n-3$; (2) Every vertex in $Q_{n,k}-V^{\prime }$ is incident to at least two fault-free vertices. This results generalize the results of the faulted hypercube $F{Q}_n$, which is a special case of $Q_{n,k}$, and have improved the theoretical evidence of the fact that $Q_{n,k}$ has excellent node-fault-tolerance when used as a topology of large-scale computer networks, thus remarkably improving the performance of the interconnect networks.

Given a finite non-empty sequence $S$ of integers, write it as $X{Y}^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. The notion of curling number of graphs has been introduced in terms of their degree sequences, analogous to the curling number of integer sequences. In this paper, we study the curling number of certain graph classes and graphs associated to given graph classes.

In this paper, we investigate and obtain the properties of higher-order Daehee sequences by using generating functions. In particular, by means of the method of coefficients and generating functions, we establish some identities involving higher-order Daehee numbers, generalized Cauchy numbers, Lah numbers, Stirling numbers of the first kind, unsigned Stirling numbers of the first kind, generalized harmonic polynomials and the numbers $P(r,n,k)$.