arXiv subjects

Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 181 records · Page 10Linked to original sources

arXiv2012

On a relation between the Szeged index and the Wiener index for bipartite graphs

{\small The Wiener index $W(G)$ of a graph $G$ is the sum of the distances between all pairs of vertices in the graph. The Szeged index $Sz(G)$ of a graph $G$ is defined as $Sz(G)=\sum_{e=uv \in E}n_u(e)n_v(e)$ where $n_u(e)$ and $n_v(e)$ are, respectively, 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 $u$. Hansen used the computer programm AutoGraphiX and made the following conjecture about the Szeged index and the Wiener index for a bipartite connected graph $G$ with $n \geq 4$ vertices and $m \geq n$ edges: $$ Sz(G)-W(G) \geq 4n-8. $$ Moreover the bound is best possible as shown by the graph composed of a cycle on 4 vertices $C_4$ and a tree $T$ on $n-3$ vertices sharing a single vertex. This paper is to give a confirmative proof to this conjecture.

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arXiv2012

A sharp upper bound for the rainbow 2-connection number of 2-connected graphs

A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is defined to be the minimum number of colors required to color the edges of $G$ such that every two distinct vertices of $G$ are connected by at least $k$ internally disjoint rainbow paths. Fujita et. al. proposed a problem that what is the minimum constant $α>0$ such that for all 2-connected graphs $G$ on $n$ vertices, we have $rc_2(G)\leq αn$. In this paper, we prove that $α=1$ and $rc_2(G)=n$ if and only if $G$ is a cycle of order $n$, settling down this problem.

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arXiv2012

Note on minimally $k$-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil $. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.

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arXiv2012

Graphs with large generalized 3-connectivity

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i, j$, where $1 \leq i, j \leq r$. For an integer $k$ with $2 \leq k \leq n$, the generalized $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $κ_2(G)$ is the connectivity of $G$. In this paper, sharp upper and lower bounds of $κ_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1 \leq κ_3(G) \leq n - 2$. Graphs of order $n$ such that $κ_3(G) = n - 2, n - 3$ are characterized, respectively.

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arXiv2012

The strong rainbow vertex-connection of graphs

A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.

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arXiv2011

Oriented diameter and rainbow connection number of a graph

The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, where $rad(G)$ is the radius of $G$ and $η(G)$ is the smallest integer number such that every edge of $G$ is contained in a cycle of length at most $η(G)$. We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph $G$ in terms of the minimum degree of $G$.

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arXiv2011

Rainbow connections for planar graphs and line graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. It was proved that computing $rc(G)$ is an NP-Hard problem, as well as that even deciding whether a graph has $rc(G)=2$ is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph is a planar bipartite graph. We also give upper bounds of the rainbow connection number of outerplanar graphs with small diameters. A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. It is known that deciding whether a given vertex-colored graph is rainbow vertex-connected is NP-Complete. We will prove that it is still NP-Complete even when the vertex-colored graph is a line graph.

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arXiv2011

Rainbow vertex-connection number of 2-connected graphs

The {\em rainbow vertex-connection number}, $rvc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its vertices such that every pair of vertices is connected by at least one path whose internal vertices have distinct colors. In this paper we first determine the rainbow vertex-connection number of cycle $C_n$ of order $n\geq 3$, and then, based on it, prove that for any 2-connected graph $G$, $rvc(G)\leq rvc(C_n)$, giving a tight upper bound for the rainbow vertex-connection. As a consequence, we show that for a connected graph $G$ with a block decomposition $B_1, B_2, ..., B_k$ and $t$ cut vertices, $rvc(G)\leq rvc(B_1)+rvc(B_2)+ ... +rvc(B_k)+t$.

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arXiv2011

Rainbow connection number of dense graphs

An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.

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arXiv2011

Note on two results on the rainbow connection number of graphs

An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. Caro et al. showed an upper bound $rc(G)\leq n-δ$ for a connected graph $G$ of order $n$ with minimum degree $δ$ in "On rainbow connection, Electron. J. Combin. 15(2008), R57". Recently, Shiermeyer gave it a generalization that $rc(G)\leq n- \frac{σ_2} 2$ in "Bounds for the rainbow connection number of graphs, Discuss. Math Graph Theory 31(2011), 387--395", where $σ_2$ is the minimum degree-sum. The proofs of both results are almost the same, both fix the minimum degree $δ$ and then use induction on $n$. This short note points out that this proof technique does not work rigorously. Fortunately, Caro et al's result is still true but under our improved proof. However, we do not know if Shiermeyer's result still hold.

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arXiv2011

Upper bound for the rainbow connection number of bridgeless graphs with diameter 3

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every pair of distinct vertices of $G$ is connected by a rainbow path. It is known that for every integer $k\geq 2$ deciding if a graph $G$ has $rc(G)\leq k$ is NP-Hard, and a graph $G$ with $rc(G)\leq k$ has diameter $diam(G)\leq k$. In foregoing papers, we showed that a bridgeless graph with diameter 2 has rainbow connection number at most 5. In this paper, we prove that a bridgeless graph with diameter 3 has rainbow connection number at most 9. We also prove that for any bridgeless graph $G$ with radius $r$, if every edge of $G$ is contained in a triangle, then $rc(G)\leq 3r$. As an application, we get that for any graph $G$ with minimum degree at least 3, $rc(L(G))\leq 3 rad(L(G))\leq 3 (rad(G)+1)$.

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arXiv2011

Further hardness results on the rainbow vertex-connection number of graphs

A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard. We also show that for any fixed integer $k\geq 2$, this problem belongs to NP-class, and so it becomes NP-Complete.

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arXiv2011

Note on rainbow connection number of dense graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. Following an idea of Caro et al., in this paper we also investigate the rainbow connection number of dense graphs. We show that for $k\geq 2$, if $G$ is a non-complete graph of order $n$ with minimum degree $δ(G)\geq \frac{n}{2}-1+log_{k}{n}$, or minimum degree-sum $σ_{2}(G)\geq n-2+2log_{k}{n}$, then $rc(G)\leq k$; if $G$ is a graph of order $n$ with diameter 2 and $δ(G)\geq 2(1+log_{\frac{k^{2}}{3k-2}}{k})log_{k}{n}$, then $rc(G)\leq k$. We also show that if $G$ is a non-complete bipartite graph of order $n$ and any two vertices in the same vertex class have at least $2log_{\frac{k^{2}}{3k-2}}{k}log_{k}{n}$ common neighbors in the other class, then $rc(G)\leq k$.

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arXiv2011

Note on the oriented diameter of graphs with diameter 3

In 1978, Chvátal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter $d$ has an orientation with diameter at most $f(d)$. For $d=3$, they proved that $8\leq f(d)\leq 24$. Until recently, Kwok, Liu and West improved the above bounds by proving $9\leq f(3)\leq 11$ in [P.K. Kwok, Q. Liu and D.B. West, Oriented diameter of graphs with diameter 3, J. Combin. Theory Ser.B 100(2010), 265-274]. In this paper, we determine the oriented diameter among the bridgeless graphs with diameter 3 that have minimum number of edges.

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arXiv2011

Note on the complexity of deciding the rainbow connectedness for bipartite graphs

A path in an edge-colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge-colored graph is (strongly) rainbow connected if there exists a rainbow (geodesic) path between every pair of vertices. The (strong) rainbow connection number of $G$, denoted by ($scr(G)$, respectively) $rc(G)$, is the smallest number of colors that are needed in order to make $G$ (strongly) rainbow connected. Though for a general graph $G$ it is NP-Complete to decide whether $rc(G)=2$, in this paper, we show that the problem becomes easy when $G$ is a bipartite graph. Moreover, it is known that deciding whether a given edge-colored (with an unbound number of colors) graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph is bipartite. We also show that a few NP-hard problems on rainbow connection are indeed NP-Complete.

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arXiv2011

On a question on graphs with rainbow connection number 2

For a connected graph $G$, the \emph{rainbow connection number $rc(G)$} of a graph $G$ was introduced by Chartrand et al. In "Chakraborty et al., Hardness and algorithms for rainbow connection, J. Combin. Optim. 21(2011), 330--347", Chakraborty et al. proved that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-Complete, and they left 4 open questions at the end, the last one of which is the following: Suppose that we are given a graph $G$ for which we are told that $rc(G)=2$. Can we rainbow-color it in polynomial time with $o(n)$ colors ? In this paper, we settle down this question by showing a stronger result that for any graph $G$ with $rc(G)=2$, we can rainbow-color $G$ in polynomial time by at most 5 colors.

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arXiv2011

Sharp upper bound for the rainbow connection number of a graph with diameter 2

Let $G$ be a connected graph. The \emph{rainbow connection number $rc(G)$} of a graph $G$ was recently introduced by Chartrand et al. Li et al. proved that for every bridgeless graph $G$ with diameter 2, $rc(G)\leq 5$. They gave examples for which $rc(G)\leq 4$. However, they could not show that the upper bound 5 is sharp. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we use different way to obtain the same upper bound, and moreover, examples are given to show that the upper is best possible.

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arXiv2011

Note on the minimal size of a graph with generalized connectivity kappa_3= 2

The concept of generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$ was introduced by Chartrand et al. in recent years. In our early paper, extremal theory for this graph parameter was started. We determined the minimal number of edges of a graph of order $n$ with $κ_{3}= 2$, i.e., for a graph $G$ of order $n$ and size $e(G)$ with $κ_{3}(G)= 2$, we proved that $e(G)\geq (6/5)n$, and the lower bound is sharp by constructing a class of graphs, only for $n\equiv 0 \ (mod \ 5)$ and $n\neq 10$. In this paper, we improve the lower bound to $\lceil(6/5)n\rceil$. Moreover, we show that for all $n\geq 4$ but $n= 9, 10$, there always exists a graph of order $n$ with $κ_{3}= 2$ whose size attains the lower bound $\lceil(6/5)n\rceil$. Whereas for $n= 9, 10$ we give examples to show that $\lceil(6/5)n\rceil+1$ is the best possible lower bound. This gives a clear picture on the minimal size of a graph of order $n$ with generalized connectivity $κ_{3}= 2$.

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