Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A vertex labeling \(f: V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(x,y) = f(x) + f(y)\), for each edge \((x,y) \in E(G)\). For each \(i \in \mathbb{Z}_2\), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|\). A vertex labeling \(f\) of a graph \(G\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined as \(\{|v_f(1) – v_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). The full friendly index set of the graph \(G\), denoted by \(FFI(G)\), is defined as \(\{|e_f(1) – e_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). In this paper, we determine \(FFI(G)\) and \(FI(G)\) for a class of cubic graphs which are twisted products of Möbius.

Let \(k\) be a non-negative integer. Two digraphs \(G = (V, A)\) and \(G’ = (V, A’)\) are \(\{k\}\)-hypomorphic if for all \(k\)-element subsets \(K\) of \(V\), the subdigraphs \(G[K]\) and \(G'[K]\) induced on \(K\) are isomorphic. The equivalence relation \(\mathcal{D}_{G,G’}\) on \(V\) is defined by: \(x \mathcal{D}_{G,G’} y\) if \(x = y\) or there exists a sequence \(x_0 = x, \ldots, x_n = y\) of elements of \(V\) satisfying \((x_i, x_{i+1}) \in A\) if and only if \((x_i, x_{i+1}) \in A’\), for all \(i\), \(0 < i k + 6\). If \(G\) and \(G’\) are two digraphs, \(\{4\}\)-hypomorphic and \(\{v – k\}\)-hypomorphic on the same vertex set \(V\) of \(uv\) vertices, and \(C\) is an equivalence class of the equivalence relation \(\mathcal{D}_{G,G’}\), then \(G'[C \setminus A]\) and \(G[C \setminus A]\) are isomorphic for all subsets \(A\) of \(V\) of at most \(k\) vertices. In particular, \(G'[C]\) and \(G[C]\) are \(\{v – k – h\}\)-hypomorphic for all \(h \in \{1, 2, \ldots, k\}\), and \(G'[C]\) and \(G[C]\) (resp. \(G’\) and \(G\)) are isomorphic. In particular, for \(k = 1\) and \(k = 4\) we obtain the result of G. Lopez and C. Rauzy [7]. As an application of the main result, we have: If \(G\) and \(G’\) are \(\{v – 4\}\)-hypomorphic on the same vertex set \(V\) of \(v > 10\) vertices, then \(G[X]\) and \(G'[X]\) are isomorphic for all subsets \(X\) of \(V\); the particular case of tournaments was obtained by Y. Boudabbous [2].

We prove that each graph in two infinite families is fixed uniquely by just two of its maximal induced subgraphs, with each of which the degree of the missing vertex is also given. One of these families contains all separable self-complementary graphs and a self-complementary graph of diameter \(3\) and order \(n\) for each \(n \geq 5\) such that \(n \equiv 0\) or \(1 \pmod{4}\). The other contains a Hamiltonian self-complementary graph of diameter \(2\) and order \(n\) for each admissible \(n \geq 8\).

Restricted strong partially balanced \(t\)-designs were first formulated by Pei, Li, Wang, and Safavi-Naini in their investigation of authentication codes with arbitration. We recently proved that optimal splitting authentication codes that are multi-fold perfect against spoofing can be characterized in terms of restricted strong partially balanced \(t\)-designs. This article investigates the existence of optimal restricted strong partially balanced 2-designs, ORSPBD\((v, 2 \times 4, 1)\), and shows that there exists an ORSPBD\((v, 2 \times 4, 1)\) for even \(v\). As its application, we obtain a new infinite class of 2-fold perfect \(4\)-splitting authentication codes.

The \(3\)-consecutive vertex coloring number \(\psi_{3c}(G)\) of a graph \(G\) is the maximum number of colors permitted in a coloring of the vertices of \(G\) such that the middle vertex of any path \(P_3\) in \(G\) has the same color as one of the ends of that \(P_3\). This coloring constraint exactly means that no \(P_3\) subgraph of \(G\) is properly colored in the classical sense. The \(3\)-consecutive edge coloring number \(\psi’_{3c}\) is the maximum number of colors permitted in a coloring of the edges of \(G\) such that the middle edge of any sequence of three edges (in a path \(P_3\) or cycle \(C_3\)) has the same color as one of the other two edges. For graphs \(G\) of minimum degree at least \(2\), denoting by \(L(G)\) the line graph of \(G\), we prove that there is a bijection between the \(3\)-consecutive vertex colorings of \(G\) and the \(3\)-consecutive edge colorings of \(L(G)\), which keeps the number of colors unchanged, too. This implies that \(\psi_{3c} = \psi’_{3c}(L(G))\); i.e., the situation is just the opposite of what one would expect at first sight.