We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph $G$, we are interested in $b_2(G)$, the minimum number of rounds until the information has spread to all vertices of graph $G$. We are also interested in finding $t_2(G)$, the minimum number of sources necessary so that the information spreads to all vertices of $G$ in $b_2(G)$ rounds. In addition to general results, we find $b_2(G)$ and $t_2(G)$ for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.

An hourglass ${\mathrm{\Gamma }}_0$ is the graph with degree sequence $\{4,2,2,2,2\}$. In this paper, for integers $j\ge i\ge 1$, the bull $B_{i,j}$ is the graph obtained by attaching endvertices of two disjoint paths of lengths $i,j$ to two vertices of a triangle. We show that every 3-connected $\{K_{1,3},{\mathrm{\Gamma }}_0,X\}$-free graph, where $X\in \{B_{2,12},B_{4,10},B_{6,8}\}$, is Hamilton-connected. Moreover, we give an example to show the sharpness of our result, and complete the characterization of forbidden induced bulls implying Hamilton-connectedness of a 3-connected {claw, hourglass, bull}-free graph.

Let $G=(V,E)$ be a simple connected graph with vertex set $V$ and edge set $E$. The Randić index of graph $G$ is the value $R(G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d(u)d(v)}}$, where $d(u)$ and $d(v)$ refer to the degree of the vertices $u$ and $v$. We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

In this paper, it is pointed out that the definition of `Fibonacci $(p,r)$-cube’ in many papers (denoted by $I{\mathrm{\Gamma }}_n^{(p,r)}$) is incorrect. The graph $I{\mathrm{\Gamma }}_n^{(p,r)}$ is not the same as the original one (denoted by $O{\mathrm{\Gamma }}_n^{(p,r)}$) introduced by Egiazarian and Astola. First, it is shown that $I{\mathrm{\Gamma }}_n^{(p,r)}$ and $O{\mathrm{\Gamma }}_n^{(p,r)}$ have different recursive structure. Then, it is proven that all the graphs $O{\mathrm{\Gamma }}_n^{(p,r)}$ are partial cubes. However, only a small part of graphs $I{\mathrm{\Gamma }}_n^{(p,r)}$ are partial cubes. It is also shown that $I{\mathrm{\Gamma }}_n^{(p,r)}$ and $O{\mathrm{\Gamma }}_n^{(p,r)}$ have different medianicity. Finally, several questions are listed for further investigation.

A $q$-total coloring of $G$ is an assignment of $q$ colors to the vertices and edges of $G$, so that adjacent or incident elements have different colors. The Total Coloring Conjecture (TCC) asserts that a total coloring of a graph $G$ has at least $\mathrm{\Delta }+1$ and at most $\mathrm{\Delta }+2$ colors. In this paper, we determine that all members of new infinite families of snarks obtained by the Kochol superposition of Goldberg and Loupekine with Blowup and Semiblowup snarks are Type~1. These results contribute to a question posed by Brinkmann, Preissmann and D. Sasaki (2015) by presenting negative evidence about the existence of Type~2 cubic graphs with girth at least 5.

In this note, we establish six Gallai theorems involving twelve minority and majority parameters. Accordingly, the complexity problems corresponding to some of these parameters are obtained.

A $k$-tree is a graph that can be formed by starting with $K_{k+1}$ and iterating the operation of making a new vertex adjacent to all the vertices of a $k$-clique of the existing graph. A structural characterization of 3-trees with diameter at most 2 is proven. This implies a corollary for planar 3-trees which leads to a description of their degree sequences.

In this paper, we present a new combinatorial characterization of Hermitian cones in $\mathrm{P}\mathrm{G}(3,q^2)$.

In this paper we consider some new weighted and alternating weighted generalized Fibonomial sums and the corresponding $q-$forms. A generalized form of weight sequences which contains squares in subscripts is discussed for the first time in the literature. The main key to get success in sums is an ability to change one sum into another that is simpler in some way. Thus, in order to prove these sums by doing some manipulations and tricks, our approach is to use classical $q-$analysis, in particular a formula of Rothe, a version of Cauchy binomial theorem and Gauss identity.

A new series of four-associate class partially balanced incomplete block designs in two replications has been proposed. The blocks of these designs are of two different sizes. The blocks can be divided into two groups such that every treatment appears in each group exactly once, and any two blocks belonging to two different groups have a constant number of treatments in common, i.e., these designs are affine resolvable.

Let $0<k\in \mathbb{Z}$. Let the star 2-set transposition graph $S{T}_k^2$ be the $(2k-1)$-regular graph whose vertices are the $2k$-strings on $k$ symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such $2k$-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph $P{C}_k^2$ has the same vertex set of $S{T}_k^2$ and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of $S{T}_k^2$. For $1<k\in \mathbb{Z}$, we show that $S{T}_k^2$ and $P{C}_k^2$, among other intermediate transposition graphs, have total colorings via $2k-1$ colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of $S{T}_k^2$ and $P{C}_k^2$, and partitions into $2k-1$ such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

This paper investigates the Turan-like problem for ${\mathcal{K}}_{r+1}^{-}$-free $(r\ge 2)$ unbalanced signed graphs, where ${\mathcal{K}}_{r+1}^{-}$ is the set of unbalanced signed complete graphs with $r+1$ vertices. The maximum number of edges and the maximum index for ${\mathcal{K}}_{r+1}^{-}$-free unbalanced signed graphs are given. Moreover, the extremal ${\mathcal{K}}_{r+1}^{-}$-free unbalanced signed graphs with the maximum index are characterized.

In this paper, we give a classification of all Mengerian $4$-uniform hypergraphs derived from graphs.

The $n$-dimensional Möbius cube $M{Q}_n$ is an important variant of the hypercube $Q_n$, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of $M{Q}_n$, and shows that if $M{Q}_n$ ($n\ge 5$) contains at most $n-2$ faulty vertices and/or edges then, for any fault-free edge $uv$ in $M{Q}_n^i(i=0,1)$ and any integer $\ell$ with $7-i⩽\ell ⩽2^n–f_v$, there is a fault-free cycle of length $\ell$ containing the edge $uv$, where $f_v$ is the number of faulty vertices. The result is optimal in some senses.

In a recent paper Cameron, Lakshmanan and Ajith [6] began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.

Let $G$ be a graph, the zero forcing number $Z(G)$ is the minimum of $|Z|$ over all zero forcing sets $Z\subseteq V(G)$. In this paper, we are interested in studying the zero forcing number of quartic circulant graphs $C_p(s,t)$, where $p$ is an odd prime. Based on the fact that $C_p(s,t)\cong C_p(1,q)$, we give the exact values of the zero forcing number of some specific quartic circulant graphs.