arXiv subjects

Hui Lei

Publications and source records attributed to Hui Lei.

At least 19 recordsLinked to original sources

arXiv2026

The Complexity of Mixed Arc-Disjoint Spanning Subdigraphs with Antistrong Connectivity

A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68--90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

cs.DM↗

arXiv2026

Feedback-arc robustness in random orientations of pseudorandom triangle-free graphs

For an oriented graph $D$, let $\vec{\alpha}(D)$ be the maximum order of an induced acyclic subdigraph, $\vec{\chi}(D)$ its dichromatic number, and $\mathrm{fas}(D)$ the minimum number of arcs whose deletion makes $D$ acyclic. We prove that for every fixed $\zeta \in (0, 1/2)$, there are triangle-free graphs $G_n$ on $n$ vertices such that a uniformly random orientation $D_n$ satisfies, $$ \left( \frac{1}{2} - \zeta \right) e(G_n[U]) < \mathrm{fas}(D_n[U]) \leq \frac{1}{2} e(G_n[U]) $$ with probability at least $1-\exp\!\left[-\Omega_\zeta\!\left(\sqrt n\,(\log n)^{3/2}\right)\right]$ simultaneously for every vertex set $U$ of size at least $C_\zeta\sqrt{n\log n}$. The upper bound is universal, so the feedback-arc ratio can be made arbitrarily close to the largest possible value, uniformly over all sufficiently large induced subdigraphs. In particular, $\vec{\alpha}(D_n) = O(\sqrt{n \log n})$, and every linear-size induced subdigraph has dichromatic number $\Omega(\sqrt{n/\log n})$. This yields $\vec{\alpha}(n) = \Theta(\sqrt{n \log n})$ and $\vec{t}(n) = \Theta\left(\sqrt{\frac{n}{\log n}}\right)$, where $\vec{\alpha}(n)$ and $\vec{t}(n)$ denote, respectively, the minimum of $\vec{\alpha}(D)$ and the maximum of $\vec{\chi}(D)$ over all oriented triangle-free graphs $D$ of order $n$. This confirms two conjectures of Aboulker, Havet, Pirot, and Schabanel.

math.CO↗

arXiv2026

Reconfiguration graphs of $K_{2,3}$-minor-free graphs

The $\ell$-reconfiguration graph of a graph $G$, denoted by $\mathcal{R}_{\ell}(G)$, is the graph whose vertices are the proper $\ell$-colorings of $G$, with an edge between two colorings if they differ in color on exactly one vertex. For any graph $G$ of treewidth at most $2$, Bousquet and Perarnau showed that $\mathcal{R}_\ell(G)$ has linear diameter for $\ell\geq 6$. This result was later extended by Bartier, Bousquet, and Heinrich, who proved that $\mathcal{R}_5(G)$ also has linear diameter. In this paper, we show that for each $\ell\geq 5$, the $\ell$-reconfiguration graphs of $K_{2,3}$-minor-free graphs, some of which include graphs of treewidth $3$, have linear diameter. As a key step in our proof, we also establish that the $(\ell-1)$-reconfiguration graphs of cactus graphs have linear diameter.

math.CO↗

arXiv2024

The Minimum Weighting Ratio Problem and Its Application in Chordal Graphs

Constructing the maximum spanning tree $T$ of an edge-weighted connected graph $G$ is one of the important research topics in computer science and optimization, and the related research results have played an active role in practical applications. In this paper, we are concerned with the ratio of the weighted sum of a spanning tree $T$ of $G$ to the weighted sum of $G$, which we try to minimize. We propose an interesting theorem to simplify this problem and show that this optimal problem can be solved in polynomial time. Furthermore, we apply the optimal problem in chordal graphs.

math.CO↗

arXiv2024

Reconfiguration graphs for vertex colorings of $P_5$-free graphs

For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any $2$-chromatic $P_5$-free graph $G$, $\mathcal{R}_k(G)$ is connected for each $k\geq 3$. On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p$, such that $\mathcal{R}_{8p}(G)$ is disconnected. In this paper, we offer a detailed classification of the connectivity of $\mathcal{R} _k(G) $ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t=3$, and $t\geq4$ with $t+1\leq k \leq {t\choose2}$. We demonstrate that $\mathcal{R}_k(G)$ remains connected for each $3$-chromatic $P_5$-free graph $G$ and each $k \geq 4$. Furthermore, for each $t\geq4$ and $t+1 \leq k \leq {t\choose2}$, we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.

math.CO↗

arXiv2024

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.

math.CO↗

arXiv2024

Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.

math.CO↗

arXiv2024

On the $k$-anti-traceability Conjecture

An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this conjecture holds when $k\leq 4$. We also show that every sufficiently large $k$-anti-traceable oriented graph admits an anti-path that contains $n-o(n)$ vertices.

math.CO↗

arXiv2023

Graph operations and a unified method for kinds of Tur\'an-type problems on paths, cycles and matchings

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for any $u,v$, where $G_{uv}$ is the graph obtained by applying Kelmans operation from $u$ to $v$; (II) $\mathcal{P}(G) <\mathcal{P}(G+e)$ for any edge $e\notin E(G)$. Let $P_k$ be a path of order $k$, $\mathcal{C}_{\geq k}$ the set of all cycles of length at least $k$ and $M_{k+1}$ a matching containing $k+1$ independent edges. In this paper, we mainly prove the following three results: (i) Let $n\geq k\geq 5$ and let $t=\left\lfloor\frac{k-1}{2}\right\rfloor$. Let $G$ be a $2$-connected $n$-vertex $\mathcal{C}_{\geq k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^1_{n,k}=\{W_{n,k,s}=K_{s}\vee ((n-k+s)K_1\cup K_{k-2s}): 2\leq s\leq t\}$. (ii) Let $n\geq k\geq 4$ and let $t=\left\lfloor\frac{k}{2}\right\rfloor-1$. Let $G$ be a connected $n$-vertex $P_{k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^2_{n,k}=\{W_{n,k-1,s}=K_{s}\vee ((n-k+s+1)K_1\cup K_{k-2s-1}): 1\leq s\leq t\}.$ (iii) Let $G$ be a connected $n$-vertex $M_{k+1}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\cong K_n$ when $n=2k+1$ and $G\in \mathcal{G}^3_{n,k}=\{K_s\vee ((n-2k+s-1)K_1\cup K_{2k-2s+1}):1\leq s\leq k\}$ when $n\geq 2k+2$. Directly derived from these three main results, we obtain a series of applications in Tur\'an-type problems, generalized Tur\'an-type problems, powers of graph degrees in extremal graph theory, and problems related to spectral radius, and signless Laplacian spectral radius in spectral graph theory.

math.CO↗

arXiv2023

Mapping tidal flat topography using time-series Sentinel-2 images and ICESat-2 data: A case study in Cixi City

Tidal flat topography provides crucial insights for understanding tidal flats and their dynamic evolution. However, the wide-ranging and rapidly changing nature of tidal flats, which are periodically submerged in shallow water, pose challenges for many current monitoring methods in terms of both efficiency and precision. In this study, we considered the dynamic process of tidal flat submergence and utilized time-series Sentinel-2 images on Google Earth Engine (GEE) to calculate the tidal flat exposure frequency. This information was used to determine the spatial extent of the tidal flats, and subsequently, by employing ICESat-2 data, we established a 1D-linear regression model based on elevation and frequency values, which realizes the inversion of the tidal flat elevation within Cixi City. The study shows the following: (1) the tidal flat exposure frequency and ICESat-2 elevation data exhibit a strong positive correlation (R2=0.85); (2) the tidal flat area within Cixi City is 115.81 km2, and the overall accuracy is 95.36%; and (3) the elevation range of the tidal flats in the study area is between -0.42 and 2.73 m, and the mean absolute error (MAE) is 0.24 m. Additionally, we consider that the temporal resolution of remote sensing imagery plays a crucial role in determining the accuracy of the elevation inversion, and we found that higher tidal flats exhibit better inversion accuracy than lower tidal flats.

physics.ao-ph↗

arXiv2022

Some Mader-perfect graph classes

The dichromatic number of $D$, denoted by $\overrightarrow{\chi}(D)$, is the smallest integer $k$ such that $D$ admits an acyclic $k$-coloring. We use $mader_{\overrightarrow{\chi}}(F)$ to denote the smallest integer $k$ such that if $\overrightarrow{\chi}(D)\ge k$, then $D$ contains a subdivision of $F$. A digraph $F$ is called Mader-perfect if for every subdigraph $F'$ of $F$, ${\rm mader }_{\overrightarrow{\chi}}(F')=|V(F')|$. We extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szab\'{o} [Dichromatic number and forced subdivisions, {\it J. Comb. Theory, Ser. B} {\bf 153} (2022) 1--30]. We also show that if $K$ is a proper subdigraph of $\overleftrightarrow{C_4}$ except for the digraph obtained from $\overleftrightarrow{C_4}$ by deleting an arbitrary arc, then $K$ is Mader-perfect.

math.CO↗

arXiv2022

Weak-odd chromatic index of special digraph classes

Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $\varphi:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd number of occurrences on each non-empty semi-cut of $v$. We call the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2 colors, use $def(D)$ to denote the defect of $D$, the minimum number of vertices in $D$ at which the above condition is not satisfied. In this paper, we give a descriptive characterization about the weak-odd chromatic index and the defect of semicomplete digraphs and extended tournaments, which generalize results of tournaments to broader classes. And we initiated the study of weak-odd edge covering on digraphs.

math.CO↗

arXiv2021

On critical graphs for the chromatic edge-stability number

The {\em chromatic edge-stability number} $es_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em $(3,2)$-critical} if $χ(G)=3$, $es_χ(G)=2$ and for any edge $e\in E(G)$, $es_χ(G-e)<es_χ(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].

math.CO↗

arXiv2021

A characterization of 4-$\chi_S$-vertex-critical graphs for packing sequences with $s_1 =1$ and $s_2\ge 3$

If $S=(s_1,s_2,\ldots)$ is a non-decreasing sequence of positive integers, then the $S$-packing $k$-coloring of a graph $G$ is a mapping $c: V(G)\rightarrow[k]$ such that if $c(u)=c(v)=i$ for $u\neq v\in V(G)$, then $d_G(u,v)>s_i$. The $S$-packing chromatic number of $G$ is the smallest integer $k$ such that $G$ admits an $S$-packing $k$-coloring. A graph $G$ is $\chi_S$-vertex-critical if $\chi_S(G-u) < \chi_S(G)$ for each $u\in V(G)$. If $G$ is $\chi_S$-vertex-critical and $\chi_S(G) = k$, then $G$ is $k$-$\chi_S$-vertex-critical. In this paper, $4$-$\chi_S$-vertex-critical graphs are characterized for sequences $S = (1,s_2, s_3, \ldots)$ with $s_2 \ge 3$. There are $28$ sporadic examples and two infinite families of such graphs.

math.CO↗

arXiv2020

A survey on star edge-coloring of graphs

The star chromatic index of a multigraph $G$, denoted $χ'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining the star chromatic index, present the interesting proofs and techniques, and collect many open problems and conjectures.

math.CO↗

arXiv2020

Some extremal results on the chromatic-stability index

The $χ$-stability index ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J.\ Combin.\ 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($k\le 5$) graphs $G$ with ${\rm es}_χ(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with $χ(G)=3$ for which ${\rm es}_χ(G)+{\rm es}_χ(\overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${\rm es}_χ(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $n\equiv 2 \pmod 3$ and $χ(G)=3$.

math.CO↗

arXiv2020

k-Ary spanning trees contained in tournaments

A rooted tree is called a $k$-ary tree, if all non-leaf vertices have exactly $k$ children, except possibly one non-leaf vertex has at most $k-1$ children. Denote by $h(k)$ the minimum integer such that every tournament of order at least $h(k)$ contains a $k$-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that $h(1)=1$. Lu et al. [J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of $h(k)$, and showed that $h(2)=4$ and $h(3)=8$. The exact values of $h(k)$ remain unknown for $k\geq 4$. A result of Erdős on the domination number of tournaments implies $h(k)=Ω(k\log k)$. In this paper, we prove that $h(4)=10$ and $h(5)\geq13$.

math.CO↗

arXiv2019

The saturation number of $K_{3,3}$

A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. Determining the saturation number of complete partite graphs is one of the most important problems in the study of saturation number. The value of $sat(n,K_{2,2})$ was shown to be $\lfloor\frac{3n-5}{2}\rfloor$ by Ollmann, and a shorter proof was later given by Tuza. For $K_{2,3}$, there has been a series of study aiming to determine $sat(n,K_{2,3})$ over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that $sat(n, K_{2,3})= 2n-3$ for all $n\geq 5$. In this paper, we prove a conjecture of Pikhurko and Schmitt that $sat(n, K_{3,3})=3n-9$ when $n \geq 9$.

math.CO↗