Discrete Applied Mathematics最新文献

A bisection of a graph is a bipartition of its vertex set in which the two parts differ in size by at most 1, and its size is the number of edges which across the two parts. Let $G$ be a graph with $n$ vertices, $m$ edges and degree sequence $d_1,d_2,\dots ,d_n$. Motivated by a few classical results on Max-Cut of graphs, Lin and Zeng proved that if $G$ is $\{C_4,C_6\}$-free and has a perfect matching, then $G$ has a bisection of size at least $m/2+\Omega \left({\sum }_{i=1}^n\sqrt{d_i}\right)$, and conjectured the same bound holds for $C_4$-free graphs with perfect matchings. In this paper, we confirm the conjecture under the additional condition that $G$ is $C_5$-free.

A bipartite graph $G$ is two-disjoint-cycle-cover edge $[r_1,r_2]$-bipancyclic if, for any vertex-disjoint edges $uv$ and $xy$ in $G$ and any even integer $\ell$ satisfying $r_1⩽\ell ⩽r_2$, there exist vertex-disjoint cycles $C_1$ and $C_2$ such that $\left|V\left(C_1\right)\right|=\ell$, $\left|V\left(C_2\right)\right|=\left|V\left(G\right)\right|-\ell$, $uv\in E\left(C_1\right)$ and $xy\in E\left(C_2\right)$. In this paper, we prove that the $n$-star graph $S_n$ is two-disjoint-cycle-cover edge $[6,\frac{n!}{2}]$-bipancyclic for $n⩾5$, and thus it is two-disjoint-cycle-cover vertex $[6,\frac{n!}{2}]$-bipancyclic for $n⩾5$. Additionally, it is examined that $S_n$ is two-disjoint-cycle-cover $[6,\frac{n!}{2}]$-bipancyclic for $n⩾4$.

The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs.

The inverse sum indeg ($ISI$) index has attracted more and more attentions, because of its significant applications in chemistry. A basic problem in the study of this topological index is the characterization trees with maximal $ISI$ value. Let $T$ be such a tree of order $n\ge 20$. Recently, Lin et al. (2022) claimed that $T$ has no vertices of degree 2. However, errors were found in their proofs. Since this result is quite important, we give a correction to the proof. Furthermore, we extend the result by proving that $T$ has no vertices of degree 2 or 3 if $n\ge 58$.

An edge-coloring of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance 2 in $G$ or in a common triangle. The injective chromatic index of $G$, ${\chi }_{inj}^{\prime }\left(G\right)$, is the minimum number of colors needed for an injective edge-coloring of $G$. In this paper, we prove that if $G$ is graph with $\Delta \left(G\right)=4$ and maximum average degree is less than $\frac{5}{2}$ (resp. $\frac{13}{5}$, $\frac{36}{13}$), then ${\chi }_{inj}^{\prime }\left(G\right)\le 6$ (resp. 7, 8).

The generalized $k$-connectivity of a graph $G$, denoted by ${\kappa }_k\left(G\right)$, is the minimum number of internally disjoint $S$-trees for any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. An $n$-dimensional burnt pancake graph $B{P}_n$ is a Cayley graph which possesses many desirable properties. In this paper, we try to evaluate the reliability of $B{P}_n$ by investigating its generalized 4-connectivity. By introducing the definition of inclusive tree and by studying structural properties of $B{P}_n$, we show that ${\kappa }_4\left(B{P}_n\right)=n-1$ for $n\ge 2$, that is, for any four vertices in $B{P}_n$, there exist ($n-1$) internally disjoint trees connecting them in $B{P}_n$.

Non-normal Boolean bent functions are one of the least understood classes of bent functions, and only a few difficult-to-find examples of such functions are known. In this paper, we consider the following two open problems on the normality of bent functions:

1. Do non-normal bent functions in 8 variables and degree 4 exist?

2. Do non-normal bent functions in the ${\mathrm{PS}}^{-}\setminus {\mathrm{PS}}_{ap}$ class exist?

We solve both of these problems by finding among the known $\mathrm{PS}$ bent functions in $n=8$ variables a non-normal bent function in the ${\mathrm{PS}}^{-}\setminus {\mathrm{PS}}_{ap}$ class.

We investigate a proper arc colouring of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals of integers. Oriented graphs having such colourings are called interval colourable. We analyse the parameter $\mathrm{icr}\left(D\right)$, which denotes the minimum number of arcs of $D$ that should be reversed so that a resulting oriented graph is interval colourable. We prove that for each non-negative integer $p$ there exists an oriented graph $D$ with the property $p\le \mathrm{icr}\left(D\right)\le 4p+1$. We show that $\mathrm{icr}\left(D\right)$ is not monotone with respect to taking subdigraphs. We give an upper bound on $\mathrm{icr}\left(D\right)$ if $D$ is an arbitrary oriented graph with finite parameter $\mathrm{icr}\left(D\right)$, and next if $D$ is any orientation of a 2-degenerate graph. Also the exact values of $\mathrm{icr}\left(D\right)$ for all orientations $D$ of some generalized Hertz graphs and generalized Sevastjanov rosettes are given. Based on special results concerning the still open problem of finding the largest possible transitive subtournament in a tournament (posed by Erdős and Moser in 1964) we refine the upper bound on $\mathrm{icr}\left(D\right)$ if $D$ is a tournament.

An $\{A,B,C,\dots \}$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{A,B,C,\dots \}$. Let $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $D\left(G\right)$ denotes the diagonal matrix of vertex degrees of $G$ and $A\left(G\right)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha }\left(G\right)$ is called the $A_{\alpha }$-spectral radius of $G$. In this paper, we explore the connections between the eigenvalues and the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph. We derive a tight sufficient condition involving the $A_{\alpha }$-spectral radius to ensure the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph, which generalizes the result on $\alpha =0$ obtained by Chen, Lv and Li [8]. Moreover, we present a tight distance signless Laplacian spectral radius condition for the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph.

If a graph $G$ contains two spanning trees $T_1,T_2$ such that for each two distinct vertices $x,y$ of $G,$ the $(x,y)$-path in each $T_i$ has no common edge and no common vertex except for the two ends, then $T_1,T_2$ are called two completely independent spanning trees (CISTs) of $G,i\in \{1,2\}.$ There are several results on the existence of two CISTs. In this paper, we prove that every 2-connected {claw, $Z_2$}-free graph with minimum degree at least 4 contains two CISTs. The bound of the minimum degree in our result is best possible.