Let \( H \) be a hypergraph of order \( n_H = |V(H)| \) and size \( m_H = |E(H)| \). The transversal number \( \tau(H) \) of a hypergraph \( H \) is the minimum number of vertices that intersect every edge of \( H \). A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A \( k \)-uniform hypergraph has all edges of size \( k \). For \( k \geq 2 \), let \( \mathcal{L}_k \) denote the class of \( k \)-uniform linear hypergraphs. We consider the problem of determining the best possible constants \( q_k \) (which depends only on \( k \)) such that \( \tau(H) \leq q_k(n_H + m_H) \) for all \( H \in \mathcal{L}_k \). It is known that \( q_2 = \frac{1}{3} \) and \( q_3 = \frac{1}{4} \). In this paper we show that \( q_4 = \frac{1}{5} \), which is better than for non-linear hypergraphs. Using the affine plane \( AG(2,4) \) of order 4, we show there are a large number of densities of hypergraphs \( H \in \mathcal{L}_4 \) such that \( \tau(H) = \frac{1}{5} (n_H + m_H) \).

Let \( F \) be a (possibly improper) edge-coloring of a graph \( G \); a vertex coloring of \( G \) is adapted to \( F \) if no color appears at the same time on an edge and on its two endpoints. If for some integer \( k \), a graph \( G \) is such that given any list assignment \( L \) to the vertices of \( G \), with \( |L(v)| \geq k \) for all \( v \), and any edge-coloring \( F \) of \( G \), \( G \) admits a coloring \( c \) adapted to \( F \) where \( c(v) \in L(v) \) for all \( v \), then \( G \) is said to be adaptably k-choosable. A \((k,d)\)-list assignment for a graph \( G \) is a map that assigns to each vertex \( v \) a list \( L(v) \) of at least \( k \) colors such that \( |L(x) \cap L(y)| \leq d \) whenever \( x \) and \( y \) are adjacent. A graph is \((k,d)\)-choosable if for every \((k,d)\)-list assignment \( L \) there is an \( L \)-coloring of \( G \). It has been conjectured that planar graphs are \((3,1)\)-choosable. We give some progress on this conjecture by giving sufficient conditions for a planar graph to be adaptably 3-choosable. Since \((k,1)\)-choosability is a special case of adaptable \( k \)-choosability, this implies that a planar graph satisfying these conditions is \((3,1)\)-choosable.

A broadcast on a nontrivial connected graph \( G = (V,E) \) is a function \( f : V \to \{0,1,\dots,\operatorname{diam}(G)\} \) such that \( f(v) \leq e(v) \) (the eccentricity of \( v \)) for all \( v \in V \). The weight of \( f \) is \( \sigma(f) = \sum_{v \in V} f(v) \). A vertex \( u \) hears \( f \) from \( v \) if \( f(v) > 0 \) and \( d(u,v) \leq f(v) \). A broadcast \( f \) is independent, or hearing independent, if no vertex \( u \) with \( f(u) > 0 \) hears \( f \) from any other vertex \( v \). We define a different type of independent broadcast, namely a boundary independent broadcast, as a broadcast \( f \) such that, if a vertex \( w \) hears \( f \) from vertices \( v_1, \dots, v_k \), \( k \geq 2 \), then \( d(w,v_i) = f(v_i) \) for each \( i \). The maximum weights of a hearing independent broadcast and a boundary independent broadcast are the \textit{hearing independence broadcast number} \( \alpha_h(G) \) and the boundary independence broadcast number \( \alpha_{bn}(G) \), respectively.

We prove that \( \alpha_{bn}(G) = \alpha(G) \) (the independence number) for any 2-connected bipartite graph \( G \) and that \( \alpha_{bn}(G) \leq n – 1 \) for all graphs \( G \) of order \( n \), characterizing graphs for which equality holds. We compare \( \alpha_{bn} \) and \( \alpha_h \) and prove that although the difference \( \alpha_h – \alpha_{bn} \) can be arbitrary, the ratio is bounded, namely \( \alpha_h / \alpha_{bn} < 2 \), which is asymptotically best possible. We deduce that \( \alpha_h(G) \leq 2n – 5 \) for all connected graphs \( G \neq P_n \) of order \( n \), which improves an existing upper bound for \( \alpha_h(G) \) when \( \alpha(G) \geq n/2 \).

We continue our studies of burn-off chip-firing games from Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121–132; MR3040546] and Australas. J. Combin. 68 (2017), no. 3, 330–345; MR3656659]. The latter article introduced randomness by choosing successive seeds uniformly from the vertex set of a graph \( G \). The length of a game is the number of vertices that fire (by sending a chip to each neighbor and annihilating one chip) as an excited chip configuration passes to a relaxed state. This article determines the probability distribution of the game length in a long sequence of burn-off games. Our main results give exact counts for the number of pairs \( (C, v) \), with \( C \) a relaxed legal configuration and \( v \) a seed, corresponding to each possible length. In support, we give our own proof of the well-known equicardinality of the set \( R \) of relaxed legal configurations on \( G \) and the set of spanning trees in the cone \( G^* \) of \( G \). We present an algorithmic, bijective proof of this correspondence.

We study homomorphism properties of signed \( K_4 \)-minor-free graphs. On the one hand, we give a necessary and sufficient condition for a signed graph \( B \) to admit a homomorphism from any signed \( K_4 \)-minor-free graph and we determine the smallest of all such signed graphs. On the other hand, we characterize the minimal cores that do not belong to the class of signed \( K_4 \)-minor-free graphs. This, in particular, gives a classification of odd-\( K_4 \)’s that are cores. Furthermore, we show some applications of our work.

Game coloring is a two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. An “eternal” version of game coloring is introduced in this paper in which the vertices are colored and re-colored from over a sequence of rounds. In a given round, each vertex is colored, or re-colored, once, so that a proper coloring is maintained. Player 1 wants to maintain a proper coloring forever, while player 2 wants to force the coloring process to fail. The eternal game chromatic number of a graph \( G \) is defined to be the minimum number of colors needed in the color set so that player 1 can always win the game on \( G \). The goal of this paper is to introduce this problem, consider several variations of this game, show its behavior on some elementary classes of graphs, and make some conjectures.

Let \( G \) be a graph with vertex set \( V \) and no isolated vertices. A subset \( S \subseteq V \) is a semipaired dominating set of \( G \) if every vertex in \( V \setminus S \) is adjacent to a vertex in \( S \) and \( S \) can be partitioned into two-element subsets such that the vertices in each subset are at most distance two apart. We present a method of building trees having a unique minimum semipaired dominating set.

We discuss the connections tying Laplacian matrices to abstract duality and planarity of graphs.

A set \( S \subseteq V(G) \) of a connected graph \( G \) is a co-secure dominating set, if \( S \) is a dominating set and for each \( u \in S \), there exists a vertex \( v \in V(G) \setminus S \), such that \( v \in N(u) \) and \( (S \setminus \{u\}) \cup \{v\} \) is a dominating set of \( G \). The minimum cardinality of the co-secure dominating set in a graph \( G \) is the co-secure domination number, \( \gamma_{cs}(G) \). In this paper, we characterise the Mycielski graphs with co-secure domination 2 and 3. We also obtained a sharp upper bound for \( \gamma_{cs}(G) \).

Given an n-connected binary matroid, we obtain a necessary and sufficient condition for its single-element coextensions to be \(n\)-connected.