arXiv subjects

Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 73 records · Page 4Linked to original sources

arXiv2018

Conflict-free connections: algorithm and complexity

A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if there is a conflict-free path between each pair of distinct vertices. We call the graph $G$ \emph{strongly conflict-free connected }if there exists a conflict-free path of length $d_G(u,v)$ for every two vertices $u,v\in V(G)$. And the \emph{strong conflict-free connection number} of a connected graph $G$, denoted by $scfc(G)$, is defined as the smallest number of colors that are required to make $G$ strongly conflict-free connected. In this paper, we first investigate the question: Given a connected graph $G$ and a coloring $c: E(or\ V)\rightarrow \{1,2,\cdots,k\} \ (k\geq 1)$ of the graph, determine whether or not $G$ is, respectively, conflict-free connected, vertex-conflict-free connected, strongly conflict-free connected under coloring $c$. We solve this question by providing polynomial-time algorithms. We then show that it is NP-complete to decide whether there is a k-edge-coloring $(k\geq 2)$ of $G$ such that all pairs $(u,v)\in P \ (P\subset V\times V)$ are strongly conflict-free connected. Finally, we prove that the problem of deciding whether $scfc(G)\leq k$ $(k\geq 2)$ for a given graph $G$ is NP-complete.

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arXiv2018

Conflict-free connection number of random graphs

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. In this paper, we show that almost all graphs have the conflict-free connection number 2. More precisely, let $G(n,p)$ denote the Erdős-Rényi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independent from other pairs. We prove that for sufficiently large $n$, $cfc(G(n,p))\le 2$ if $p\ge\frac{\log n +α(n)}{n}$, where $α(n)\rightarrow \infty$. This means that as soon as $G(n,p)$ becomes connected with high probability, $cfc(G(n,p))\le 2$.

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arXiv2018

More on limited packings in graphs

A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain the tight Nordhaus-Gaddum-type result of this parameter for general $k$. At last, we investigate the relationship among the open packing number, the packing number and $2$-limited packing number of trees.

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arXiv2018

Conflict-free connection of trees

We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2} n\rceil$, which completely confirms the conjecture of Li and Wu. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with $2^{k-1}$ vertices is $k-1$. At last, we study trees which are $cfc$-critical, and prove that if a tree $T$ is $cfc$-critical, then the conflict-free connection coloring of $T$ is equivalent to the edge ranking of $T$.

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arXiv2018

Graphs with conflict-free connection number two

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. For a graph $G,$ let $C(G)$ be the subgraph of $G$ induced by its set of cut-edges. In this paper, we first show that, if $G$ is a connected non-complete graph $G$ of order $n\geq 9$ with $C(G)$ being a linear forest and with the minimum degree %$δ(G)\geq 2$, then $cfc(G)=2$ for $4 \leq n\leq 8 $; if $δ(G)\geq \max\{3, \frac{n-4}{5}\}$, then $cfc(G)=2$. The bound on the minimum degree is best possible. Next, we prove that, if $G$ is a connected non-complete graph of order $n\geq 33$ with $C(G)$ being a linear forest and with $d(x)+d(y)\geq \frac{2n-9}{5}$ for each pair of two nonadjacent vertices $x, y$ of $V(G)$, then $cfc(G)=2$. Both bounds, on the order $n$ and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on $G$ and the number of cut edges in $G$.

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arXiv2017

Non-jumping Numbers for 5-Uniform Hypergraphs

Let $\ell$ and $r$ be integers. A real number $α\in [0,1)$ is a jump for $r$ if for any $\varepsilon > 0$ and any integer $m,\ m \geq r$, any $r$-uniform graph with $n > n_0(\varepsilon,m)$ vertices and at least α+ \varepsilon)\binom{n}{r}$ edges contains a subgraph with $m$ vertices and at least $(α+c)\binom{m}{r}$ edges, where $c=c(α)$ does not depend on $\varepsilon$ and $m$. It follows from a theorem of Erdős, Stone and Simonovits that every $α\in [0,1)$ is a jump for $r=2$. Erdős asked whether the same is true for $r \geq 3$. However, Frankl and Rödl gave a negative answer by showing that $1-\frac{1}{\ell^{r-1}}$ is not a jump for $r$ if $r \geq 3$ and $\ell >2r$. Peng gave more sequences of non-jumping numbers for $r=4$ and $r\geq 3$. However, there are also a lot of unknowns on determining whether a number is a jump for $r \geq 3$. Following a similar approach as that of Frankl and Rödl, we give several sequences of non-jumping numbers for $r=5$, and extend one of the results to every $r \geq 5$, which generalize the above results.

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arXiv2017

Extremality of graph entropy based on degrees of uniform hypergraphs with few edges

Let $\mathcal{H}$ be a hypergraph with $n$ vertices. Suppose that $d_1,d_2,\ldots,d_n$ are degrees of the vertices of $\mathcal{H}$. The $t$-th graph entropy based on degrees of $\mathcal{H}$ is defined as $$ I_d^t(\mathcal{H}) =-\sum_{i=1}^{n}\left(\frac{d_i^{t}}{\sum_{j=1}^{n}d_j^{t}}\log\frac{d_i^{t}}{\sum_{j=1}^{n}d_j^{t}}\right) =\log\left(\sum_{i=1}^{n}d_i^{t}\right)-\sum_{i=1}^{n}\left(\frac{d_i^{t}}{\sum_{j=1}^{n}d_j^{t}}\log d_i^{t}\right), $$ where $t$ is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of $I_d^t(\mathcal{H})$ for $t=1$, when $\mathcal{H}$ is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.

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arXiv2017

A Survey on Monochromatic Connections of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.

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arXiv2017

Nordhaus-Gaddum-type theorem for conflict-free connection number of graphs

An edge-colored graph $G$ is \emph{conflict-free connected} if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is defined as the smallest number of colors that are needed in order to make $G$ conflict-free connected. In this paper, we determine all trees $T$ of order $n$ for which $cfc(T)=n-t$, where $t\geq 1$ and $n\geq 2t+2 $. Then we prove that $1\leq cfc(G)\leq n-1$ for a connected graph $G$, and characterize the graphs $G$ with $cfc(G)=1,n-4,n-3,n-2,n-1$, respectively. Finally, we get the Nordhaus-Gaddum-type theorem for the conflict-free connection number of graphs, and prove that if $G$ and $\overline{G}$ are connected, then $4\leq cfc(G)+cfc(\overline{G})\leq n$ and $4\leq cfc(G)\cdot cfc(\overline{G})\leq2(n-2)$, and moreover, $cfc(G)+cfc(\overline{G})=n$ or $cfc(G)\cdot cfc(\overline{G})=2(n-2)$ if and only if one of $G$ and $\overline{G}$ is a tree with maximum degree $n-2$ or a $P_5$, and the lower bounds are sharp.

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arXiv2017

Conflict-free vertex-connections of graphs

A path in a vertex-colored graph is called \emph{conflict free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be \emph{conflict-free vertex-connected} if any two vertices of the graph are connected by a conflict-free path. This paper investigates the question: For a connected graph $G$, what is the smallest number of colors needed in a vertex-coloring of $G$ in order to make $G$ conflict-free vertex-connected. As a result, we get that the answer is easy for $2$-connected graphs, and very difficult for connected graphs with more cut-vertices, including trees.

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arXiv2017

Conflict-free connection numbers of line graphs

A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is a conflict-free path connecting them. For a connected graph $G$, the \emph{conflict-free connection number} of $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required to make $G$ conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We first show that for an arbitrary connected graph $G$, there exists a positive integer $k$ such that $cfc(L^k(G))\leq 2$. Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph $G$ and an arbitrary positive integer $k$, we always have $cfc(L^{k+1}(G))\leq cfc(L^k(G))$, with only the exception that $G$ is isomorphic to a star of order at least~$5$ and $k=1$. Finally, we obtain the exact values of $cfc(L^k(G))$, and use them as an efficient tool to get the smallest nonnegative integer $k_0$ such that $cfc(L^{k_0}(G))=2$.

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arXiv2017

On various (strong) rainbow connection numbers of graphs

An edge-coloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow path. Similarly, the \emph{strong rainbow connection number} $src(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al.~in 2008. Subsequently, vertex-coloured versions of both parameters, $rvc(G)$ and $srvc(G)$, and a total-coloured version of the rainbow connection number, $trc(G)$, were introduced. In this paper we introduce the strong total rainbow connection number $strc(G)$, which is the version of the strong rainbow connection number using total-colourings. Among our results, we will determine the strong total rainbow connection numbers of some special graphs. We will also compare the six parameters, by considering how close and how far apart they can be from one another. In particular, we will characterise all pairs of positive integers $a$ and $b$ such that, there exists a graph $G$ with $trc(G)=a$ and $strc(G)=b$, and similarly for the functions $rvc$ and $srvc$.

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arXiv2017

Tight Nordhaus-Gaddum-type upper bound for total-rainbow connection number of graphs

A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$, denoted by $trc(G)$, is the minimum number of colors required in a total-coloring of $G$ to make $G$ total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus-Gaddum-type upper bound for the total-rainbow connection number. We prove that if $G$ and $\overline{G}$ are connected complementary graphs on $n$ vertices, then $trc(G)+trc(\overline{G})\leq 2n$ when $n\geq 6$ and $trc(G)+trc(\overline{G})\leq 2n+1$ when $n=5$. Examples are given to show that the upper bounds are sharp for $n\geq 5$. This completely solves a conjecture in [Y. Ma, Total rainbow connection number and complementary graph, Results in Mathematics 70(1-2)(2016), 173-182].

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arXiv2017

More on the $k$-color connection number of a graph

An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors needed to color the edges of $G$ so that $G$ is $k$-color connected. First, we prove that let $H$ be a subdivision of a connected graph $G$, then $cc_{k}(H)\leq cc_{k}(G)$. Second, we give sufficient conditions to guarantee that $cc_{k}(G)=k$ in terms of minimum degree and the number of edges for 2-connected graphs. As a byproduct, we show that almost all graphs have the $k$-color connection number $k$. At last, we investigate the relationship between the $k$-color connection number and the rainbow connection number for a connected graph. In addition, we give exact values of $k$-color connection numbers for some graph classes: subdivisions of the wheel and the complete graph, and the generalised $θ$-graph.

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arXiv2017

Constructions of graphs and trees with partially prescribed spectrum

It is shown how a connected graph and a tree with partially prescribed spectrum can be constructed. These constructions are based on a recent result of Salez that every totally real algebraic integer is an eigenvalue of a tree. Our result implies that for any (not necessarily connected) graph $G$, there is a tree $T$ such that the characteristic polynomial $P(G,x)$ of $G$ can divide the characteristic polynomial $P(T,x)$ of $T$, i.e., $P(G,x)$ is a divisor of $P(T,x)$.

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arXiv2016

Erdős-Gallai-type results for total monochromatic connection of graphs

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. In this paper, we study two kinds of Erdős-Gallai-type problems for $tmc(G)$ and completely solve them.

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arXiv2016

The $k$-proper index of complete bipartite and complete multipartite graphs

Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if no two adjacent edges of it are assigned the same color. Let $k$ be a fixed integer with $2\leq k\leq n$. For a vertex subset $S\subseteq V(G)$ with $|S|\geq 2$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$ . A \emph{$k$-proper coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a proper $S$-tree $T$ in $G$. The minimum number of colors that are needed in a $k$-proper coloring of $G$ is defined as the \emph{$k$-proper index} of $G$, denoted by $px_k(G)$. In this paper, we determine the 3-proper index of all complete bipartite and complete multipartite graphs and partially determine the $k$-proper index of them for $k\geq 4$.

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arXiv2016

Nordhaus-Gaddum-type theorem for total proper connection number of graphs

A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A path $P$ in a total-colored graph $G$ is called a \emph{total-proper path} if $(i)$ any two adjacent edges of $P$ are assigned distinct colors; $(ii)$ any two adjacent internal vertices of $P$ are assigned distinct colors; $(iii)$ any internal vertex of $P$ is assigned a distinct color from its incident edges of $P$. The total-colored graph $G$ is \emph{total-proper connected} if any two distinct vertices of $G$ are connected by a total-proper path. The \emph{total-proper connection number} of a connected graph $G$, denoted by $tpc(G)$, is the minimum number of colors that are required to make $G$ total-proper connected. In this paper, we first characterize the graphs $G$ on $n$ vertices with $tpc(G)=n-1$. Based on this, we obtain a Nordhaus-Gaddum-type result for total-proper connection number. We prove that if $G$ and $\overline{G}$ are connected complementary graphs on $n$ vertices, then $6\leq tpc(G)+tpc(\overline{G})\leq n+2$. Examples are given to show that the lower bound is sharp for $n\geq 4$. The upper bound is reached for $n\geq 5$ if and only if $G$ or $\overline{G}$ is the tree with maximum degree $n-2$.

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