On a Vizing-type integer domination conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $\gamma_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Bre\v{s}ar, Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{\{k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{2\}$-domination number, $\gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{\{2\}}(G\square H)\ge \gamma_{R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.Comment: 8 page

Advancements in Research Mathematics through AI: A Framework for Conjecturing

This paper introduces a general framework for computer-based conjecture generation, particularly those conjectures that mathematicians might deem substantial and elegant. We describe our approach and demonstrate its effectiveness by providing examples of its application in producing publishable research and unexpected mathematical insights. We anticipate that our discussion of computer-assisted mathematical conjecturing will catalyze further research into this area and encourage the development of more advanced techniques than the ones presented herein

Upper bounds on the k-forcing number of a graph

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most $k$ non-colored neighbors, then each of its non-colored neighbors becomes colored. When $k=1$, this is equivalent to the zero forcing number, usually denoted with $Z(G)$, a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the $k$-forcing number. Notable among these, we show that if $G$ is a graph with order $n \ge 2$ and maximum degree $\Delta \ge k$, then $F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k + 1 +\min{\{\delta,k\}}}$. This simplifies to, for the zero forcing number case of $k=1$, $Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}$. Moreover, when $\Delta \ge 2$ and the graph is $k$-connected, we prove that $F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}$, which is an improvement when $k\leq 2$, and specializes to, for the zero forcing number case, $Z(G)= F_1(G) \le \frac{(\Delta -2)n+2}{\Delta -1}$. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the $k$-forcing number and the connected $k$-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure