The subject matter for this paper is GDDs with three groups of sizes \(n_1,n(n\geq n_1)\) and \(n+1\), for \(n_1=1\, or\, 2\) 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 \(n_1 = 1\, o\,r 2\) 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 \(n_1=1\, and \,2\), 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 \(n_1 + n_2 + n_3=0,4\) \((mod\, 12)\) 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 \(K{s,t}\). For a positive integer s and two bipartite graphs G and H, the s-bipartite Ramsey number \(BR_s (G, H)\) 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 \(BR_s (K_{2,3}, K_{3,3})\) for all possible \(s\). More precisely, we show that \(BR_s (K_{2,3}, K_{3,3})=13\) for \(s\) \(\in\) {8,9}, \(BR_s (K_{2,3}, K_{3,3})=12\) for \(s \in \{10,11\}\), \(BR_s (K_{2,3}, K_{3,3})=10\) for \(s = 12\), \(BR_s (K_{2,3}, K_{3,3})=8\) for \(s \in \{13,14\}\), \(BR_s (K_{2,3}, K_{3,3})=6\) for \(s \in \{15,16,…, 20\}\), and \(BR_s (K_{2,3}, K_{3,3})=4\) 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 \(q^2- q + 1 – \bar{t}\bar{z}\), for every pair \((\bar{t},\bar{z}) \in P_1 \cup P_2\), where \(P_1\) and \(P_2\) are the pair sets \(P_1=\{(t,z)\in\mathbb{Z}\times\mathbb{Z}:\frac{q}{2}-2\leq t\leq q-3,0\leq z\leq\frac{q}{2}-2\}\) and \(P_2=\{(t,z)\in\mathbb{Z}\times\mathbb{Z}:0\leq t\leq \frac{q}{2}-3,0\leq z\leq{q}-1\}\), for \(q \geq 8\).

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 \(G\, \underrightarrow{v}(a_1,…,a_s)\) means that for every coloring of the vertices of G in s colors there exists \(i \in {1,…, s}\) such that there is a monochromatic \(a_i\)-clique of color i. The vertex Folkman numbers \(F_v(a_1, …, a_s; q)\) are defined by the equality:
\[F_v(a_1, …, a_s; q) = min\{|V(G)| : G\, \underrightarrow{v}(a_1,…,a_s)\,\, and K_q \nsubseteq G\}\].
Let \(m = \sum^s_{i=1} (a_i – 1) + 1\). It is easy to see that \(F_v(a_1,…, a_s; q) = m\) if \(q \geq m + 1\). In [11] it is proved that \(F_v(a_1, …, a_s;m) = m +max {a_1,…, a_s}\). We know all the numbers \(F_v(a_1,…, a_s; m – 1)\) when \(max {a_1,..,a_s} ≤ 5\) and none of these numbers is known if \(max{a_1, …, a_s} ≥ 6\). 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 \(F_v(a_1, …, a_s;m – 1)\) when \(max(a_1, …, a_s) = 7\).

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 \(spc_k(G)\) 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 \(spc_2(K_{r,s}) = \left\lceil ^{r-1}\sqrt{s} \right\rceil\) for 2 ≤ r ≤ s.

Suppose \(G = (V, E)\) is a simple graph and \(f: (V\cup E) → {1,2,…, k}\) 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 \(C(u) \neq C(v)\) for every \(uv \in E(G)\). The minimum k for which such a chromatic number of G, and is denoted by \(X_at (G)\). 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 \(X_at (G) ≤ 6\) 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, \(\sum_{u \in N_{(v)}}f(u)\geq2\). 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 \(\gamma_dR(G)/2 ≤ \gamma_I(G) ≤ 2\gamma_dR(G)/3\), and characterize all trees T with \(\gamma_I(T) = 2\gamma_dR (T)/3\).