Discrete Applied Mathematics最新文献

A path in an edge-coloured graph is called properly coloured path, or more simply, proper path if every two consecutive edges receive distinct colours. An edge-coloured graph $G$ is called properly $k$-connected if every two vertices are connected by at least $k$ internally pairwise vertex-disjoint proper paths. The proper $k$-connection number of a $k$-connected graph $G$, denoted by $p{c}_k\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly $k$-connected. In this paper, we study the proper 2-connection number $p{c}_2\left(G\right)$ of graphs. We prove a new upper bound for $p{c}_2\left(G\right)$ and determine several classes of graphs satisfying $p{c}_2\left(G\right)=2$. Among these are all graphs satisfying the Chvátal and Erdős condition ($\alpha \left(G\right)\le \kappa \left(G\right)$) with two exceptions). We also study the relationship between the proper 2-connection number $p{c}_2\left(G\right)$ and the proper connection number $pc\left(G\right)$ of the Cartesian product of two connected graphs.

Let $G$ be a connected graph. The weighted Mostar index of $G$ is defined as $w^{+}Mo\left(G\right)={\sum }_{e=uv\in E\left(G\right)}(d_u+d_v)|n_u-n_v|,$where $d_u$ and $d_v$ denote the degrees of vertex $u$ and vertex $v$, $n_u$ and $n_v$ are the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, respectively. In this paper, we determine the maximum value and the second maximum value of the weighted Mostar indices for cacti with order $n$ and $k$ cycles, and identify the extremal graphs with the maximum and the second maximum weighted Mostar indices.

Let $G=(V,E)$ be a graph. A set $S\subseteq V$ is a total restrained dominating set if every vertex is adjacent to a vertex in $S,$ and every vertex in $V-S$ is adjacent to a vertex in $V-S$. The total restrained domination number of $G$, denoted ${\gamma }_{tr}\left(G\right)$, is the smallest cardinality of a total restrained dominating set of $G$. In this paper we show that if $G$ is a $K_{1,\ell }$-free graph with $\delta \ge \ell \ge 3$ and $\delta \ge 5$, then ${\gamma }_{tr}\left(G\right)\le n\left(1-\frac{(2\delta -3)}{{\left(2\delta \right)}^{\frac{\delta }{\delta -1}}}+o_{\delta }\left(1\right)\right).$

Diagnosability is a fundamental consideration when designing an interconnected network. The PMC and MM${}^{\ast }$ fault diagnosis models are the two most commonly used models. Both the $g$-good-neighbour diagnosability and $g$-extra diagnosability of an interconnection network have been two of the hot topics in the intersectional research areas of Graph theory and Computer Science, which become increasingly attractive for new solutions to real-world problems. However, there are still some problems in the transformation from the concepts of Computer Science to that of mathematics. In this paper, we systematically study such problems and give a strict proof from concepts to mathematical conclusions. In the terms of results, we not only give the relationship between $g$-good-neighbour diagnosabilities of the network under PMC model and MM${}^{\ast }$ model, but also between $g$-extra diagnosabilities of the network under PMC and MM${}^{\ast }$ models. To apply our results, we give an application on the enhanced hypercube in the end and derive a lemma explaining whether these are 3-cycles in enhanced hypercubes and how many common neighbours for two vertices of enhanced hypercubes under different values of $k$ in the meantime.

For non-decreasing sequence of integers $(a_1,a_2,\dots ,a_k)$, an $(a_1,a_2,\dots ,a_k)$-packing coloring of $G$ is a partition of $V\left(G\right)$ into $k$ subsets $V_1,V_2,\dots ,V_k$ such that the distance between any two vertices $x,y\in V_i$ is at least $a_{i+1}$, $1\le i\le k$. Yang and Wu (2023) proved that every 3-irregular subcubic graph is $(1,1,3)$-packing colorable. We introduce here a simpler and shorter proof of this result.

In 1964, Moon extended Cayley’s formula to a nice expression of the number of spanning trees in complete graphs containing any fixed spanning forest. After nearly 60 years, Dong and the first author discovered the second Moon-type formula: an explicit formula of the number of spanning trees in complete bipartite graphs containing any fixed spanning forest. Followed this direction, Li, Chen and Yan found the Moon-type formula for complete 3- and 4-partite graphs. These are the only families of graphs that have the corresponding Moon-type formulas. In this paper, we first determine resistance distances in the vertex-weighted complete split graph $S_{m,n}^{\omega }$. Then we obtain the Moon-type formula for the vertex-weighted complete split graph $S_{m,n}^{\omega }$, that is, the weighted spanning tree enumerator of $S_{m,n}^{\omega }$ containing any fixed spanning forest.

A set $S$ of vertices in an isolate-free graph $G$ is a total dominating set if every vertex of $G$ is adjacent to some other vertex in $S$. A total coalition in $G$ consists of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a total dominating set but whose union $X\cup Y$ is a total dominating set of $G$. Such sets $X$ and $Y$ are said to form a total coalition. A total coalition partition in $G$ is a vertex partition $\Psi =\{V_1,V_2,\dots ,V_k\}$ such that for all $i\in \left[k\right]$, the set $V_i$ forms a total coalition with another set $V_j$ for some $j$, where $j\in \left[k\right]\setminus \left\{i\right\}$. The total coalition number $C_t\left(G\right)$ in $G$ equals the maximum order of a total coalition partition in $G$. It is known that if $G$ is an isolate-free graph, then $2\le C_t\left(G\right)\le n$. We characterize graphs with smallest possible total coalition number, that is, we characterize isolate-free graphs $G$ satisfying $C_t\left(G\right)=2$. Moreover we characterize graphs $G$ with $\delta \left(G\right)=1$ satisfying