A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.

Explicit expressions for all the primitive idempotents in the ring \(R_{2^n} = {F}_q[x]/(x^{2^n} – 1)\), where \(q\) is an odd prime power, are obtained. Some lower bounds on the minimum distances of the irreducible cyclic codes of length \(2^n\) over \({F}_q\) are also obtained.

In this note we prove that all connected Cayley graphs of every finite group \(Q \times H\) are \(1\)-factorizable, where \(Q\) is any non-trivial group of \(2\)-power order and \(H\) is any group of odd order.

A graph \(G\) is called super vertex-magic total labelings if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(v) + \sum_{u \sim v} f(vu) = C\), where the sum is over all vertices \(u\) adjacent to \(v\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\), \(f(E(G)) = \{|V(G)|+1,|V(G)|+2,\ldots,|V(G)|+|E(G)|\}\). \({The Knödel graphs}\) \(W_{\Delta,n}\) have even \(n \geq 2\) vertices and degree \(\Delta\), \(1 \leq \Delta \leq \lfloor\log_2 n\rfloor\). The vertices of \(W_{\Delta,n}\) are the pairs \((i,j)\) with \(i = 1,2\) and \(0 \leq i \leq n/2-1\). For every \(j\), \(0 \leq j \leq n/2-1\), there is an edge between vertex \((1,j)\) and every vertex \((2,(j+2^k-1) \mod (n/2))\), for \(k=0,\ldots,\Delta-1\). In this paper, we show that \(W_{3,n}\) is super vertex-magic for \(n \equiv 0 \mod 4\).

Evolutionary graphs were initially proposed by Lieberman \(et \;al\). and evolutionary dynamics on two levels are recently introduced by Traulsen et al. We now introduce a new type of evolutionary dynamics,evolutionary graphs on two levels, and the fixation probability is analyzed. Some interesting results, evolutionary graphs on two levels are more stable than single level evolutionary graphs, are obtained in this paper.

A vertex \(k\)-ranking of a graph \(G\) is a function \(c: V(G) \to \{1,\ldots,k\}\) such that if \(c(u) = c(v)\), \(u,v \in V(G)\), then each path connecting vertices \(u\) and \(v\) contains a vertex \(w\) with \(c(w) > c(u)\). If each vertex \(v\) has a list of integers \(L(v)\) and for a vertex ranking \(c\) it holds \(c(v) \in L(v)\) for each \(v \in V(G)\), then \(c\) is called an \(L\)-list \(k\)-ranking, where \(\mathcal{L} = \{L(v) : v \in V(G)\}\). In this paper, we investigate both vertex and edge (vertex ranking of a line graph) list ranking problems. We prove that both problems are NP-complete for several classes of acyclic graphs, like full binary trees, trees with diameter at most \(4\), and comets. The problem of finding vertex (edge) \(\mathcal{L}\)-list ranking is polynomially solvable for paths and trees with a bounded number of non-leaves, which includes trees with diameter less than \(4\).

In this paper we determine unique graph with largest spectral radius among all tricyclic graphs with \(n\) vertices and \(k\) pendant edges.

A new proof is given to the following result of ours. Let \(G\) be an outerplanar graph with maximum degree \(\Delta \geq 3\). The chromatic number \(\chi(G^2)\) of the square of \(G\) is at most \(\Delta+2\), and \(\chi(G^2) = \Delta+1\) if \(\Delta \geq 7\).

Some designs using the action of the linear fractional groups \(L_2(q)\), \(q = 11, 13, 16, 17, 19, 23\) are constructed. We will show that \(L_2(q)\) or its automorphism group acts as the full automorphism group of each of the constructed designs except in the case \(q = 16\). For designs constructed from \(L_2(16)\), we will show that \(L_2(16)\), \(L_2(16) : 2\), \(L_2(16) : 4\) or \(S_{17}\) can arise as the full automorphism group of the design.

For odd \(n \geq 5\), the Flower Snark \(F_n = (V, E)\) is a simple undirected cubic graph with \(4n\) vertices, where \(V = \{a_i : 0 \leq i \leq n-1\} \cup \{b_i : 0 \leq i \leq n-1\} \cup \{c_i : 0 \leq i \leq 2n-1\}\) and \(E = \{b_ib_{(i+1)\mod(n)}: 0 \leq i \leq n-1\} \cup \{c_ic_{(i+1)\mod(2n)} : 0 \leq i \leq 2n-1\} \cup \{a_ib_i,a_ic_i,a_ic_{n+i} : 0 \leq i \leq n-1\}\). For \(n = 3\) or even \(n \geq 4\), \(F_n\) is called the related graph of Flower Snark. We show that the crossing number of \(F_n\) equals \(n – 2\) if \(3 \leq n \leq 5\), and \(n\) if \(n \geq 6\).