Showing papers by "Xueliang Li published in 2020"

Graph colorings under global structural conditions.

Abstract: More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the {\it global chromatic numbers}, and the classic or traditional chromatic numbers the {\it local chromatic numbers}. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.

Bounds for the rainbow disconnection number of graphs

Abstract: An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which meet the upper bound of the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a conjecture that $\lambda^+(G)\leq \textnormal{rd}(G)\leq \lambda^+(G)+1$, where $\lambda^+(G)$ is the upper edge-connectivity, and prove the conjecture for many classes of graphs, to support it. Finally, we give the relationship between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number rvd$(L(G))$ of the line graph $L(G)$ of $G$.

Complexity Results for the Proper Disconnection of Graphs

Abstract: For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned by different colors. An edge-colored graph is called proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is defined as the minimum number of colors that are needed to make G proper disconnected. In this paper, we first show that it is NP-complete to decide whether a given k-edge-colored graph G with \(\varDelta (G)=4\) is proper disconnected. Then, for a graph G with \(\varDelta (G)\le 3\) we show that \(pd(G)\le 2\) and determine the graphs with \(pd(G)=1\) and 2 in polynomial time, respectively, when the set of vertices with degree 3 in G is an independent set. Finally, we show that for a general graph G, deciding whether \(pd(G)=1\) is NP-complete, even if G is bipartite.

Spectral properties of inverse sum indeg index of graphs

Abstract: The inverse sum indeg (ISI) index is a vertex-degree-based topological index that was selected by Vukicevic and Gasperov in 2010 as a significant predictor of the total surface area of octane isomers. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the ISI index from an algebraic viewpoint. We introduce suitably modified versions of the classical adjacency matrix and the Laplacian matrix involving the degrees of the vertices of a graph. Moreover, we formulate the ISI index in terms of these matrices.

Extremal graphs and classification of planar graphs by MC-numbers

Abstract: An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.

Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

Abstract: In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $\delta^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with $\delta^c(G)\geq \frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

Abstract: The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image α(x), where α is an automorphism of graph. The aim of this paper is to compute the automorphism group of some classes of cubic polyhedral graphs and then we determine their Wiener index. In addition, we investigate the GP-index of these classes of graphs.

Note on rainbow cycles in edge-colored graphs

Abstract: Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $\delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $\delta^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $\delta^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $\delta^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.

On edge-path eigenvalues of graphs

Abstract: Let G be a graph with vertex set V ( G ) = { v 1 , … , v n } and E P ( G ) be an n × n matrix whose ( i , j ) -entry is the maximum number of internally edge-disjoint paths between v i and v j , if...

Strong rainbow disconnection in graphs

Abstract: Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {\it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then the edge-cut is called a {\it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called \emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({\it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {\it strong rainbow disconnection coloring} (srd-{\it coloring} for short) of $G$. For a connected graph $G$, the \emph{strong rainbow disconnection number} (srd-{\it number} for short) of $G$, denoted by $\textnormal{srd}(G)$, is the smallest number of colors that are needed in order to make $G$ strong rainbow disconnected. In this paper, we first characterize the graphs with $m$ edges such that $\textnormal{srd}(G)=k$ for each $k \in \{1,2,m\}$, respectively, and we also show that the srd-number of a nontrivial connected graph $G$ equals the maximum srd-number among the blocks of $G$. Secondly, we study the srd-numbers for the complete $k$-partite graphs, $k$-edge-connected $k$-regular graphs and grid graphs. Finally, we show that for a connected graph $G$, to compute $\textnormal{srd}(G)$ is NP-hard. In particular, we show that it is already NP-complete to decide if $\textnormal{srd}(G)=3$ for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is strong rainbow disconnected.

Rainbow triangles in edge-colored complete graphs

Abstract: Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $\delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $\delta^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $\delta^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $\delta^c(G)\geq \frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $\delta^c(G)\geq \frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $n\geq 20$, if $\delta^c(G)\geq \frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $n\geq 8$ and $\delta^c(G)\geq \frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $n\geq 20$ to $n\geq 7$, the best possible.