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.

In this paper, we provide bounds for the crossing number of mesh connected trees and 3-regular mesh of trees.

The ordinary factorial may be written in terms of the Stirling numbers of the second kind as shown by Quaintance and Gould and the odd double factorial in terms of the Stirling numbers of the first kind as shown by Callan. During the preparation of an expository paper on Wolstenholme’s Theorem, we discovered an expression for the odd double factorial using the Stirling numbers of the second kind. This appears to be the first such identity involving both positive and negative integers and the result is outlined here.

Let \( G \) be a \( k \)-connected (\( k > 2 \)) graph of order \( n \). If \( \chi(G) > n-k \), then \( G \) is Hamiltonian or \( K_k \vee (K_{n-2k}) \) with \( n > 2k+1 \), where \( \chi(G) \) is the chromatic number of the graph \( G \).