Discrete Applied Mathematics最新文献

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e., the stretch index of $G$, denoted by $\sigma \left(G\right)$, is the goal of the $t$-admissibility problem. Split graphs are 3-admissible and can be partitioned into three subclasses: split graphs with $\sigma =1,2$ or 3. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with $\Delta$ or $\Delta +1$ colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, it is conjectured that any graph can be colored with $\Delta +1$ or $\Delta +2$ colors, and thus can be classified as Type 1 or Type 2. Both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with $\sigma =2$. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.

In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz. the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same order; for each choice of a graph type $A$ and an equivalence relation $B$, there is a graph, the $B$ super$A$ graph defined on $G$. The resulting nine graphs (of which eight were shown to be in general distinct) form a two-dimensional hierarchy. In the present paper, we consider these graphs further. We prove universality properties for the conjugacy supergraphs of various types, adding the nilpotent, solvable and enhanced power graphs to the commuting graphs considered in the rest of the paper, and also examine their relation to the invariably generating graph of the group. We also show that supergraphs can be expressed as graph compositions, in the sense of Schwenk, and use this representation to calculate their Wiener index. We illustrate these by computing Wiener index of equality supercommuting and conjugacy supercommuting graphs for dihedral and dicyclic groups.

In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a $(2/3)$-approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a $(1/2)$-approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of $\frac{3}{\Delta +2}$ of the greedy algorithm for general graphs of maximum degree $\Delta$.

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A $d$-interval is the union of $d$ disjoint intervals on the real line, and a graph is a $d$-interval graph if it is the intersection graph of $d$-intervals. In particular, it is a unit $d$-interval graph if it admits a $d$-interval representation where every interval has unit length.

Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is $\mathrm{NP}$-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also $\mathrm{NP}$-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit $d$-interval graphs for any $d\ge 2$, which does not follow directly in graph recognition problems — as an example, it took almost 20 years to close the gap between $d=2$ and $d>2$ for the recognition of $d$-track interval graphs. Our result has several implications, including that for every $d\ge 2$, recognizing $(x,\dots ,x)$$d$-interval graphs and depth $r$ unit $d$-interval graphs is $\mathrm{NP}$-complete for every $x\ge 11$ and every $r\ge 4$.

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a convex geometry. In this work, we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.

Topological indices are mathematical descriptors used in the field of chemistry to characterize the topological structure of chemical compounds. The Randić index ($R$), the geometric–arithmetic index ($GA$), and the arithmetic–geometric index ($AG$) represent three widely recognized topological indices. In most scenarios, the properties of $AG$ and $GA$ exhibit opposing tendencies. Furthermore, it is observed that, $AG\left(G\right)>R\left(G\right)$ and $GA\left(G\right)>R\left(G\right)$ for any given graph $G$. Our focus is thus directed towards investigating the gaps between $AG$ and $R$, as well as $GA$ and $R$. We find that the invariants $AG-R$ and $GA-R$ correlate well with some molecular properties. Numerous upper and lower bounds for the quantities $AG-R$ and $GA-R$ are computed for general graphs, bipartite graphs, chemical graphs, trees, and chemical trees, in terms of graph order, with an emphasis on characterizing extremal graphs.

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$. The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A\left(G\right)$ and $D\left(G\right),$ respectively. In 2017, Nikiforov proposed the $A_{\alpha }$-matrix: $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right),$ where $\alpha \in [0,1].$ The largest eigenvalue of this novel matrix is called the $A_{\alpha }$-index of $G.$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_{\alpha }$-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$-vertex graphs having the minimum $A_{\alpha }$-index with dissociation number $\tau$, where $\tau ⩾\lceil \frac{2}{3}n\rceil .$ Finally, we identify all the connected $n$-vertex graphs with dissociation number $\tau \in \{2,\lceil \frac{2}{3}n\rceil ,n-1,n-2\}$ having the minimum $A_{\alpha }$-index.

In this paper, we provide an efficient algorithm to construct almost optimal $(k,n,d)$-superimposed codes with runlength constraints. A $(k,n,d)$-superimposed code of length $t$ is a $t\times n$ binary matrix such that any two 1’s in each column are separated by a run of at least $d$ 0’s, and such that for any column $c$ and any other $k-1$ columns, there exists a row where $c$ has 1 and all the remaining $k-1$ columns have 0. These combinatorial structures were introduced by Agarwal et al. (2020), in the context of Non-Adaptive Group Testing algorithms with runlength constraints.

By using Moser and Tardos’ constructive version of the Lovász Local Lemma, we provide an efficient randomized Las Vegas algorithm of complexity $\Theta \left(t{n}^2\right)$ for the construction of $(k,n,d)$-superimposed codes of length $t=O(dklogn+k^{2}logn)$. We also show that the length of our codes is shorter, for $n$ sufficiently large, than that of the codes whose existence was proved in Agarwal et al. (2020).

A graph $G$ has the strong parity property if for every subset $X\subseteq V\left(G\right)$ with $\left|X\right|$ even, $G$ has a spanning subgraph $F$ satisfying $\delta \left(F\right)\ge 1$, $d_F\left(u\right)\equiv 1$ (mod 2) for any $u\in X$, and $d_F\left(v\right)\equiv 0$ (mod 2) for any $v\in V\left(G\right)\setminus X$. In this paper, we give a spectral radius condition to guarantee that a connected graph has the strong parity property.