Suppose \(G\) is a graph. The minimum number of paths (trees, forests, linear forests, star forests, complete bipartite graphs, respectively) needed to decompose the edges of \(G\) is called the path number (tree number, arboricity, linear arboricity, star arboricity and biclique number, respectively) of \(G\). These numbers are denoted by \(p(G), t(G), a(G), la(G), sa(G), r(G)\), respectively. For integers \(1 \leq k \leq n\), let \(C_{n,k}\) be the graph with vertex set \(\{a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\}\) and edge set \(\{a_ib_j :i=1,2,\ldots ,n,j \equiv i+1,i+2, \ldots ,i+k \text{(mod n)}\}\). We call \(C_{n,k}\) a crown. In this paper, we prove the following results:

1. \(p(C_{n,k}) = \begin{cases}
   n & \text{if \(k\) is odd}, \\
   {(\frac{k}{2})+1} & \text{if \(k\) is even}.
   \end{cases}\)
2. \(a(C_{n,k}) = t(C_{n,k}) = la(C_{n,k}) = \left\lceil \frac{k+1}{2} \right\rceil\) if \(k \geq 2\).
3. For \(k \geq 3\) and \(k \in \{3,5\} \cup \{n-3,n-2,n-1\}\),
   \[sa(C_{n,k}) = \begin{cases}
   \left\lceil \frac{k}{2} \right\rceil + 1 & \text{if \(k\) is odd}, \\
   \left\lceil \frac{k}{2} \right\rceil + 2 & \text{if \(k\) is even}.
   \end{cases}\]
4. \(r(C_{n,k}) = n\) if \(k \leq \frac{n+1}{2}\) or \(\gcd(k,n) = 1\).

Due to (3), (4), we propose the following conjectures.

\(\textbf{Conjecture A}\). For \(3 \leq k \leq n-1\),
\[sa(C_{n,k}) = \begin{cases}
\left\lceil \frac{k}{2} \right\rceil + 1 & \text{if \(k\) is odd}, \\
\left\lceil \frac{k}{2} \right\rceil + 2 & \text{if \(k\) is even}.
\end{cases}\]
\(\textbf{Conjecture B}\). For \(1 \leq k \leq n-1\), \(r(C_{n,k}) = n\).

Let \(G = (V, E)\) be a graph and \(A\) a non-trivial Abelian group, and let \(\mathcal{F}(G, A)\) denote the set of all functions \(f: E(G) \to A\). Denote by \(D\) an orientation of \(E(G)\). Then \(G\) is \(A\)-colorable if and only if for every \(f \in \mathcal{F}(G, A)\) there exists an \(A\)-coloring \(c: V(G) \to A\) such that for every \(e = (x,y) \in E(G)\) (assumed to be directed from \(x\) to \(y\)), \(c(x) – c(y) \neq f(e)\). If \(G\) is a graph, we define its group chromatic number \(\chi_1(G)\) to be the minimum number \(m\) for which \(G\) is \(A\)-colorable for any Abelian group \(A\) of order \(\geq m\) under the orientation \(D\). In this paper, we investigated the properties of the group chromatic number, proved the Brooks Type theorem for \(\chi_1(G)\), and characterized all bipartite graphs with group chromatic number at most \(3\), among other things.

A signed graph is an unoriented graph with a given partition \(E = E^+ \bigcup E^-\) of its edge-set. We define the arc signed graph \({A}(G)\) of an oriented graph \(G\) (G has no multiple arcs, opposite arcs, and loops). The arc signed graphs are similar to the line graphs. We prove both a Krausz-type characterization and a forbidden induced subgraph characterization (like the theorem of Beineke and Robertson on line graphs). Unlike line graphs, there are infinitely many minimal forbidden induced subgraphs for the arc signed graphs. Nevertheless, the arc signed graphs are polynomially recognizable. Also, we obtain a result similar to Whitney’s theorem on line graphs.

For a vertex \(v\) in a graph \(G\), we denote by \(N^2(v)\) the set \((N_1(N_1(v))\setminus \{v\})\cup N_1(v)=\{x\in V(G): 1 \leq d(x,v) \leq 2\}\), where \(d(x,v)\) denotes the distance between \(x\) and \(v\). A vertex \(v\) is \(N^2\)-locally connected if the subgraph induced by \(N^2(v)\) is connected. A graph \(G\) is called \(N^2\)-locally connected if every vertex of \(G\) is \(N^2\)-connected. A well-known result by Oberly and Sumner is that every connected locally connected claw-free graph on at least three vertices is Hamiltonian. This result was improved by Ryjacek using the concept of second-type neighborhood. In this paper, using the concept of \(N^2\)-locally connectedness, we show that every connected \(N^2\)-locally connected claw-free graph \(G\) without vertices of degree \(1\), which does not contain an induced subgraph \(H\) isomorphic to one of \(G_1, G_2, G_3\), or \(G_4\), is Hamiltonian, hereby generalizing the result of Oberly and Sumner (J. Graph Theory, \(3 (1979) 351-356\))and the result of \(Ryjacek\)( J. Graph Theory, \(14 (1990)\) 321-381)

On the gracefulness of graph \(C_m\bigcup P_n\), Frucht and Salinas that proved \(C_m\bigcup P_n\) is graceful and conjectured: \(C_m\bigcup P_n\) is graceful if and only if \(m+n=7\). In this paper, we prove graph \(C_m\bigcup P_n\) is graceful, for \(m=4k, n=k+2, k+3, 2k+1,\ldots, 2k+5;\) \(m=4k+1, n=2k, 3k+1, 4k+1;\) \(m=4k+2 n=3k, 3k+1,
4k+1; m=4k+3, n=2k+1, 3k, 4k\).

Let \(\nu(\mathbb{Z}^m)\) be the minimal number of colors enough to color the \(m\)-dimensional integer grid \(\mathbb{Z}^m\) so that there would be no infinite monochromatic symmetric subsets. Banakh and Protasov [3] compute \(\nu(\mathbb{Z}^m) = m+1\). For the one-dimensional case this just means that one can color positive integers in red, while negative integers in blue, thereby avoiding an infinite monochromatic symmetric subset by a trivial reason. This motivates the question what changes if we allow only colorings unlimited in both directions (in “all” directions for \(m > 1\)). In this paper we show that then \(\nu(\mathbb{Z})\) increases by \(1\), whereas for higher dimensions the values \(\nu(\mathbb{Z}^m)\) remain unaffected.
Furthermore we examine the density properties of a set \(A \subseteq \mathbb{Z}^m\) that ensure the existence of infinite symmetric subsets or arbitrarily large finite symmetric subsets in \(A\). In the case that \(A\) is a sequence with small gaps, we prove a multi-dimensional analogue of the Szemerédi theorem, with symmetric subsets in place of arithmetic progressions. A similar two-dimensional statement is known for collinear subsets (Pomerance [10]), whereas for two-dimensional arithmetic progressions even the corresponding version of van der Waerden’s theorem is known to be false.

The eccentricity of a vertex \(v\) in a connected graph \(G\) is the distance between \(v\) and a vertex farthest from \(v\). For a vertex \(v\), we define the edge-added eccentricity of \(v\) as the minimum eccentricity of \(v\) in all graphs \(G+e\), taken over all edges \(e\) in the complement of \(G\). A graph is said to be edge-added stable (or just stable) if the eccentricity and the edge-added eccentricity are the same for all vertices in the graph. This paper describes properties of edge-added eccentricities and edge-added stable graphs.

In this paper, we find explicit formulas or generating functions for the cardinalities of the sets \(S_n(T,\tau)\) of all permutations in \(S_n\) that avoid a pattern \(\tau \in S_k\) and a set \(T, |T| \geq 2,\) of patterns from \(S_3\). The main body of the paper is divided into three sections corresponding to the cases \(|T| = 2, 3\) and \(|T| \geq 4\). As an example, in the fifth section, we obtain the complete classification of all cardinalities of the sets \(S_n(T,\tau)\) for \(k = 4\).

The concept of weakly associative lattices (i.e. relational systems with a reflexive and antisymmetric relation \(\leq\), in which for each pair of elements there exist a least upper and a greatest lower bound) was introduced in [3] and [5]. In [4] WU-systems are defined, i.e. weakly associative lattices with the unique bound property, and their equivalence with projective planes is described. In this paper we introduce WU\(_{\lambda}\)-systems, and discuss their relation to symmetric \(2\)-\((v,k,\lambda)\) designs equipped with a special “loop-free” mapping.

It is shown in this paper that every \(2\)-connected claw-free graph containing a \(k\)-factor has a connected \([k,k+1]\)-factor, where \(k \geq 2\).