arXiv subjects

Yongtang Shi

Publications and source records attributed to Yongtang Shi.

At least 55 records · Page 3Linked to original sources

arXiv2019

The annihilation number does not bound the 2-domination number from the above

The $2$-domination number $γ_2(G)$ of a graph $G$ is the minimum cardinality of a set $S\subseteq V(G)$ such that every vertex from $V(G)\setminus S$ is adjacent to at least two vertices in $S$. The annihilation number $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of its edges. It was conjectured that $γ_2(G) \leq a(G) +1$ holds for every connected graph $G$. The conjecture was earlier confirmed, in particular, for graphs of minimum degree $3$, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the $2$-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of Jakovac from [Discrete Appl.\ Math.\ 260 (2019) 178--187].

math.CO↗

arXiv2019

Exact rainbow numbers for matchings in plane triangulations

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of its edges have different colors. Denote by $M_t$ a matching of size $t$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. Jendrol', Schiermeyer and Tu initiated to investigate the rainbow numbers for matchings in plane triangulations, and proved some bounds for the value of $rb({\mathcal {T}_n},M_t)$. Chen, Lan and Song proved that $2n+3t-14 \le rb(\mathcal {T}_n, M_t)\le 2n+4t-13$ for all $n\ge 3t-6$ and $t \ge 6$. In this paper, we determine the exact values of $rb({\mathcal {T}_n},M_t)$ for large $n$, namely, $rb({\mathcal {T}_n},M_t)=2n+3t-14$ for all $n \ge 9t+3$ and $t\ge 7$.

math.CO↗

arXiv2019

Extremal Theta-free planar graphs

Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230], that is, how many edges can an $\mathcal{F}$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,\mathcal{F})$ to be the maximum number of edges in an $\mathcal{F}$-free planar graph on $n $ vertices. Dowden obtained the tight bounds $ex_{_\mathcal{P}}(n,C_4)\leq15(n-2)/7$ for all $n\geq4$ and $ex_{_\mathcal{P}}(n,C_5)\leq(12n-33)/5$ for all $n\geq11$. In this paper, we continue to promote the idea of determining $ex_{_\mathcal{P}}(n,\mathcal{F})$ for certain classes $\mathcal{F}$. Let $Θ_k$ denote the family of Theta graphs on $k\ge4$ vertices, that is, graphs obtained from a cycle $C_k$ by adding an additional edge joining two non-consecutive vertices. The study of $ex_{_\mathcal{P}}(n,Θ_4)$ was suggested by Dowden. We show that $ex_{_\mathcal{P}}(n,Θ_4)\leq12(n-2)/5$ for all $n\geq 4$, $ex_{_\mathcal{P}}(n,Θ_5)\leq5(n-2)/2$ for all $n\ge5$, and then demonstrate that these bounds are tight, in the sense that there are infinitely many values of $n$ for which they are attained exactly. We also prove that $ex_{_\mathcal{P}}(n,C_6)\le ex_{_\mathcal{P}}(n,Θ_6)\le 18(n-2)/7$ for all $n\ge6$.

math.CO↗

arXiv2018

On the complexity of k-rainbow cycle colouring problems

An edge-coloured cycle is $rainbow$ if all edges of the cycle have distinct colours. For $k\geq 1$, let $\mathcal{F}_{k}$ denote the family of all graphs with the property that any $k$ vertices lie on a cycle. For $G\in \mathcal{F}_{k}$, a $k$-$rainbow$ $cycle$ $colouring$ of $G$ is an edge-colouring such that any $k$ vertices of $G$ lie on a rainbow cycle in $G$. The $k$-$rainbow$ $cycle$ $index$ of $G$, denoted by $crx_{k}(G)$, is the minimum number of colours needed in a $k$-rainbow cycle colouring of $G$. In this paper, we restrict our attention to the computational aspects of $k$-rainbow cycle colouring. First, we prove that the problem of deciding whether $crx_1=3$ can be solved in polynomial time, but that of deciding whether $crx_1 \leq k$ is NP-Complete, where $k\geq 4$. Then we show that the problem of deciding whether $crx_2=3$ can be solved in polynomial time, but those of deciding whether $crx_2 \leq 4$ or $5$ are NP-Complete. Furthermore, we also consider the cases of $crx_3=3$ and $crx_3 \leq 4$. Finally, We prove that the problem of deciding whether a given edge-colouring (with an unbounded number of colours) of a graph is a $k$-rainbow cycle colouring, is NP-Complete for $k=1$, $2$ and $3$, respectively. Some open problems for further study are mentioned.

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arXiv2018

Improved bounds for rainbow numbers of matchings in plane triangulations

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of whose edges have different colors. Denote by $kK_2$ a matching of size $k$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. In [S. Jendrol$'$, I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158--164], the authors determined the exact values of $rb(\mathcal {T}_n, kK_2)$ for $2\leq k \le 4$ and proved that $2n+2k-9 \le rb(\mathcal {T}_n, kK_2) \le 2n+2k-7+2\binom{2k-2}{3}$ for $k \ge 5$. In this paper, we improve the upper bounds and prove that $rb(\mathcal {T}_n, kK_2)\le 2n+6k-16$ for $n \ge 2k$ and $k\ge 5$. Especially, we show that $rb(\mathcal {T}_n, 5K_2)=2n+1$ for $n \ge 11$.

math.CO↗

arXiv2018

Gallai-Ramsey numbers of $C_{10}$ and $C_{12}$

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the Gallai-Ramsey number $GR(H_1, \ldots, H_k)$ is the least integer $n$ such that every Gallai $k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H_i$ in color $i$ for some $i \in \{1, \ldots, k\}$. When $H = H_1 = \cdots = H_k$, we simply write $GR_k(H)$. We continue to study Gallai-Ramsey numbers of even cycles and paths. For all $n\ge3$ and $k\ge1$, let $G_i=P_{2i+3}$ be a path on $2i+3$ vertices for all $i\in\{0,1, \ldots, n-2\}$ and $G_{n-1}\in\{C_{2n}, P_{2n+1}\}$. Let $ i_j\in\{0,1,\ldots, n-1\}$ for all $j\in\{1, \ldots, k\}$ with $ i_1\ge i_2\ge\cdots\ge i_k $. Song recently conjectured that $GR(G_{i_1}, \ldots, G_{i_k}) = 3+\min\{i_1, n^*-2\}+\sum_{j=1}^k i_j$, where $n^* =n$ when $G_{i_1}\ne P_{2n+1}$ and $n^* =n+1$ when $G_{i_1}= P_{2n+1}$. This conjecture has been verified to be true for $n\in\{3,4\}$ and all $k\ge1$. In this paper, we prove that the aforementioned conjecture holds for $n \in\{5, 6\}$ and all $k \ge1$. Our result implies that for all $k \ge 1$, $GR_k(C_{2n}) = GR_k(P_{2n}) = (n-1)k+n+1$ for $n\in\{5,6\}$ and $GR_k(P_{2n+1})= (n-1)k+n+2$ for $1\le n \le6 $.

math.CO↗

arXiv2018

Extremal $H$-free planar graphs

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

math.CO↗

arXiv2018

The edit distance function of some graphs

The edit distance function of a hereditary property $\mathscr{H}$ is the asymptotically largest edit distance between a graph of density $p\in[0,1]$ and $\mathscr{H}$. Denote by $P_n$ and $C_n$ the path graph of order $n$ and the cycle graph of order $n$, respectively. Let $C_{2n}^*$ be the cycle graph $C_{2n}$ with a diagonal, and $\widetilde{C_n}$ be the graph with vertex set $\{v_0, v_1, \ldots, v_{n-1}\}$ and $E(\widetilde{C_n})=E(C_n)\cup \{v_0v_2\}$. Marchant and Thomason determined the edit distance function of $C_6^{*}$. Peck studied the edit distance function of $C_n$, while Berikkyzy et al. studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of $C_8^{*}$, $\widetilde{C_n}$ and $P_n$, respectively.

math.CO↗

arXiv2018

Extremal problems on saturation for the family of $k$-edge-connected graphs

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,\mathcal{F}_k)=(k-1)n-{k\choose2}$, we prove $sat(n,\mathcal{F}'_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$. We also prove $ex(n,\mathcal{F}'_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for $\mathcal{F}_k$-saturated and $\mathcal{F}'_k$-saturated graphs.

math.CO↗

arXiv2018

Degree powers in graphs with a forbidden forest

Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Turán-type problem for $e_p(G)$: Given a positive integer $p$ and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Turán number. Caro and Yuster determined the function $ex_p(n, P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the path on $\ell$ vertices. In this paper, we generalise this result and determine $ex_p(n,F)$ for sufficiently large $n$, where $p\geq 2$ and $F$ is a linear forest. We also determine $ex_p(n,S)$, where $S$ is a star forest; and $ex_p(n,B)$, where $B$ is a broom graph with diameter at most six.

math.CO↗

arXiv2017

Planar anti-Ramsey numbers for paths and cycles

Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. Analogous to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number of $H$ is the maximum number of edges of a planar graph on $n$ vertices without containing $H$ as a subgraph. The study of $ar_{_\mathcal{P}}(n, H)$ (under the name of rainbow numbers) was initiated by Horňák, Jendrol$'$, Schiermeyer and Soták [J Graph Theory 78 (2015) 248--257]. In this paper we study planar anti-Ramsey numbers for paths and cycles. We first establish lower bounds for $ar_{_\mathcal{P}}(n, P_k)$ when $n\ge k\ge8$. We then improve the existing lower bound for $ar_{_\mathcal{P}}(n, C_k)$ when $k\geq 5$ and $n\geq k^2-k$. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for $ar_{_\mathcal{P}}(n, C_6)$ when $n\ge8$ and $ar_{_\mathcal{P}}(n, C_7)$ when $n\geq 13$, respectively.

math.CO↗

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.

math.CO↗

arXiv2017

Star chromatic index of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $χ'_s(G)\le3$ for an arbitrary graph $G$. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs $G$ with $δ(G)\le2$ such that $χ'_s(G)>k$ but $χ'_s(G-v)\le k$ for any $v\in V(G)$, where $k\in\{5,6\}$. We finally apply the structure results, along with a simple discharging method, to prove that every subcubic multigraph $G$ is star $6$-edge-colorable if $mad(G)<5/2$, and star $5$-edge-colorable if $mad(G)<24/11$, respectively, where $mad(G)$ is the maximum average degree of a multigraph $G$. This partially confirms the conjecture of Dvořák, Mohar and Šámal.

math.CO↗

arXiv2017

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.

math.CO↗

arXiv2017

The Turan number of 2P_7

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P_{k}$. Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20(2011) 837--853] determined the exact value of $ex(n,kP_\ell)$ for large values of $n$. Yuan and Zhang [The Turán number of disjoint copies of paths, Discrete Math. 340(2)(2017) 132--139] completely determined the value of $ex(n,kP_3)$ for all $n$, and also determined $ex(n,F_m)$, where $F_m$ is the disjoint union of $m$ paths containing at most one odd path. They also determined the exact value of $ex(n,P_3\cup P_{2\ell+1})$ for $n\geq 2\ell+4$. Recently, Bielak and Kieliszek [The Turán number of the graph $2P_5$, Discuss. Math. Graph Theory 36(2016) 683--694], Yuan and Zhang [Turán numbers for disjoint paths, arXiv: 1611.00981v1] independently determined the exact value of $ex(n,2P_5)$. In this paper, we show that $ex(n,2P_{7})=\max\{[n,14,7],5n-14\}$ for all $n \ge 14$, where $[n,14,7]=(5n+91+r(r-6))/2$, $n-13\equiv r\,(\text{mod }6)$ and $0\leq r< 6$.

math.CO↗

arXiv2017

Total rainbow connection of digraphs

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) total rainbow connection number of digraphs. Results on the (strong) total rainbow connection number of biorientations of graphs, tournaments and cactus digraphs are presented.

math.CO↗

arXiv2017

Quantitative Graph Theory: A new branch of graph theory and in network science

In this paper, we describe {\sc quantitative graph theory} and argue it is a new graph-theoretical branch in network science, however, with significant different features compared to classical graph theory. The main goal of quantitative graph theory is the structural quantification of information contained in complex networks by employing a {\it measurement approach} based on numerical invariants and comparisons. Furthermore, the methods as well as the networks do not need to be deterministic but can be statistic. As such this complements the field of classical graph theory, which is descriptive and deterministic in nature. We provide examples of how quantitative graph theory can be used for novel applications in the context of the overarching concept network science.

cs.SI↗

arXiv2017

Rainbow vertex connection of digraphs

An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the concept of \emph{rainbow vertex-connection}. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.

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