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Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 235 records · Page 13Linked to original sources
arXiv2009

The rainbow $k$-connectivity of two classes of graphs

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $κ$-connected graph $G$ and an integer $k$ with $1\leq k\leq κ$, the rainbow $k$-connectivity $rc_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Let $G$ be a complete $(\ell+1)$-partite graph with $\ell$ parts of size $r$ and one part of size $p$ where $0\leq p <r$ (in the case $p=0$, $G$ is a complete $\ell$-partite graph with each part of size $r$). This paper is to investigate the rainbow $k$-connectivity of $G$. We show that for every pair of integers $k\geq 2$ and $r\geq 1$, there is an integer $f(k,r)$ such that if $\ell\geq f(k,r)$, then $rc_k(G)=2$. As a consequence, we improve the upper bound of $f(k)$ from $(k+1)^2$ to $ck^{3/2}+C$, where $0<c<1$, $C=o(k^{3/2})$, and $f(k)$ is the integer such that if $n \geq f(k)$ then $rc_k(K_n)=2$.

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arXiv2009

Sharp bounds for the generalized connectivity $κ_3(G)$

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $κ(S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. A collection $\{T_1,T_2,...,T_\ell\}$ of trees in $G$ with this property is called an internally disjoint set of trees connecting $S$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ is the connectivity of $G$. In general, the investigation of $κ_k(G)$ is very difficult. We therefore focus on the investigation on $κ_3(G)$ in this paper. We study the relation between the connectivity and the 3-connectivity of a graph. First we give sharp upper and lower bounds of $κ_3(G)$ for general graphs $G$, and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if $G$ is a connected planar graph, then $κ(G)-1 \leq κ_3(G)\leq κ(G)$, and give some classes of graphs which attain the bounds. In the end we show that the problem whether $κ(G)=κ_3(G)$ for a planar graph $G$ can be solved in polynomial time.

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arXiv2009

A proof of the conjecture on hypoenergetic graphs with maximum degree $Δ\leq 3$

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. A graph $G$ of order $n$ is said to be hypoenergetic if $E(G)<n$. Majstorović et al. conjectured that complete bipartite graph $K_{2,3}$ is the only hypoenergetic connected quadrangle-containing graph with maximum degree $Δ\leq 3$. This paper is devoted to giving a confirmative proof to the conjecture.

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arXiv2009

The limiting spectral distribution of the generalized Wigner matrix

The properties of eigenvalues of large dimensional random matrices have received considerable attention. One important achievement is the existence and identification of the limiting spectral distribution of the empirical spectral distribution of eigenvalues of Wigner matrix. In the present paper, we explore the limiting spectral distribution for more general random matrices, and, furthermore, give an application to the energy of general random graphs, which generalizes the result of Nikiforov.

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arXiv2009

Note for Nikiforov's two conjectures on the energy of trees

The energy $E$ of a graph is defined to be the sum of the absolute values of its eigenvalues. Nikiforov in {\it ``V. Nikiforov, The energy of $C_4$-free graphs of bounded degree, Lin. Algebra Appl. 428(2008), 2569--2573"} proposed two conjectures concerning the energy of trees with maximum degree $Δ\leq 3$. In this short note, we show that both conjectures are true.

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arXiv2009

Hypoenergetic and strongly hypoenergetic trees

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. An $n$-vertex graph is said to be hypoenergetic if $E(G)<n$ and strongly hypoenergetic if $E(G)<n-1$. In this paper, we consider hypoenergetic and strongly hypoenergetic trees. For any given $n$ and $Δ$, the existence of both hypoenergetic and strongly hypoenergetic trees of order $n$ and maximum degree $Δ$ is completely characterized.

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arXiv2008

Long heterochromatic paths in heterochromatic triangle free graphs

In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle free graphs $G$ we show that if, for any vertex $v\in V(G)$, $d^c(v)\geq k\geq 6$, then $G$ a heterochromatic path of length at least $\frac{3k}{4}$.

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arXiv2008

Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$

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arXiv2007

Monochromatic and heterochromatic subgraph problems in a randomly colored graph

Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by $\mathcal{K}_n^r$. In the paper, we investigate the asymptotic properties of several kinds of monochromatic and heterochromatic subgraphs in $\mathcal{K}_n^r$. Accurate threshold functions in some cases are also obtained.

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arXiv2007

Nonexistence of triples of nonisomorphic connected graphs with isomorphic connected $P_3$-graphs

In the paper "Broersma and Hoede, {\it Path graphs}, J. Graph Theory {\bf 13} (1989) 427-444", the authors proposed a problem whether there is a triple of mutually nonisomorphic connected graphs which have an isomorphic connected $P_3$-graph. For a long time, this problem remains unanswered. In this paper, we give it a negative answer that there is no such triple, and thus completely solve this problem.

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arXiv2007

Complexity of the conditional colorability of graphs

For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number, denoted by $χ_r(G)$. It is easy to see that the conditional coloring is a generalization of the traditional vertex coloring for which $r=1$. In this paper, we consider the complexity of the conditional colorings of graphs. The main result is that the conditional $(3,2)$-colorability is $NP$-complete for triangle-free graphs with maximum degree at most 3, which is different from the old result that the traditional 3-colorability is polynomial solvable for graphs with maximum degree at most 3. This also implies that it is $NP$-complete to determine if a graph of maximum degree 3 is $(3,2)$- or $(4,2)$-colorable. Also we have proved that some old complexity results for traditional colorings still hold for the conditional colorings.

cs.DM↗
arXiv2007

Dynamic 3-Coloring of Claw-free Graphs

A {\it dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex of degree at least 2 in $G$ will be adjacent to vertices with at least 2 different colors. The smallest number $k$ for which a graph $G$ can have a dynamic $k$-coloring is the {\it dynamic chromatic number}, denoted by $χ_d(G)$. In this paper, we investigate the dynamic 3-colorings of claw-free graphs. First, we prove that it is $NP$-complete to determine if a claw-free graph with maximum degree 3 is dynamically 3-colorable. Second, by forbidding a kind of subgraphs, we find a reasonable subclass of claw-free graphs with maximum degree 3, for which the dynamically 3-colorable problem can be solved in linear time. Third, we give a linear time algorithm to recognize this subclass of graphs, and a linear time algorithm to determine whether it is dynamically 3-colorable. We also give a linear time algorithm to color the graphs in the subclass by 3 colors.

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arXiv2007

Rainbow number of matchings in regular bipartite graphs

Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to $G$. Denote $mK_2$ a matching of size $m$ and $B_{n,k}$ a $k$-regular bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|=n$ and $k\leq n$. In this paper we give an upper and lower bound for $rb(B_{n,k},mK_2)$, and show that for given $k$ and $m$, if $n$ is large enough, $rb(B_{n,k},mK_2)$ can reach the lower bound. We also determine the rainbow number of matchings in paths and cycles.

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arXiv2007

On the existence of a rainbow 1-factor in proper coloring of K_{rn}^{(r)}

El-Zanati et al proved that for any 1-factorization $\mathcal{F}$ of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor.

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