arXiv subjects

Xiaxia Guan

Publications and source records attributed to Xiaxia Guan.

17 recordsLinked to original sources

arXiv2026

A characterization of tight ($ k, 0 $)-stable graphs

Let k and l be two non-negative integers with k > l. A graph G is (k,l)-stable if alpha(G - S) >= alpha(G) - l for every subset S of V(G) with |S| = k, where alpha(G) denotes the independence number of G. Dong and Wu established that alpha(G) <= floor((n - k + 1)/2) + l for a (k, l)-stable graph G, where n is the order of G. A (k, l)-stable graph G is tight if alpha(G) = floor((n - k + 1)/2) + l. In this paper, we provide a complete characterization of tight (k, 0)-stable graphs for k >= 4. In particular, we prove that tight (k, 0)-stable graphs are K_{k+1} and K_{k+2} for k >= 5, which not only extends the result of Liu, Song and Wang [J. Graph Theory 110(2) (2025), 193-199] from k >= 24 to k >= 5, but also proves the conjecture of Dong and Luo [Electron. J. Comb. 32(4) (2025), 4-45] once more.

math.CO↗

arXiv2026

Connectivity for slice-projections of connected polymatroids

It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

math.CO↗

arXiv2025

A recursive definition for the polymatroid Tutte polynomial

The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently.

math.CO↗

arXiv2025

On the coefficients of interior and exterior polynomials of polymatroids

The Tutte polynomial is an important invariant of graphs and matroids. Chen and Guo \emph{[Adv. in Appl. Math. 166 (2025) 102868.]} proved that for a $(k+1)$-edge connected graph $G$ and for any $i$ with $0\leq i <\frac{3(k+1)}{2}$, $$[y^{g-i}]T_{G}(1,y)=\binom{|V(G)|+i-2}{i}-\sum_{j=0}^{i}\binom{|V(G)|+i-2-j}{i-j}|\mathcal{SC}_{j}(G)|,$$ where $g=|E(G)|-|V(G)|+1$, $\mathcal{SC}_{j}(G)$ is the set of all minimal edge cuts with $j$ edges, $T_{G}(x,y)$ is the Tutte polynomial of the graph $G$, and $[y^{g-i}]T_{G}(1,y)$ denotes the coefficient of $y^{g-i}$ in the polynomial $T_{G}(1,y)$. Recently, Ma, Guan and Jin \emph{[arXiv.2503.06095, 2025.]} generalized this result from graphs to matroids and obtained the dual result on coefficients of $T_M(x,1)$ of matroids $M$ at the same time. In 2013, as a generalization of $T_{G}(x,1)$ and $T_{G}(1,y)$ of graphs $G$ to hypergraphs, Kálmán \emph{[Adv. Math. 244 (2013) 823-873.]} introduced interior and exterior polynomials for connected hypergraphs. Chen and Guo posed a problem that can one generalize these results of graphs to interior and exterior polynomials of hypergraphs? In this paper, we solve it in the affirmative by obtaining results for more general polymatroids, which include the case of hypergraphs and also generalize the results of matroids due to Ma, Guan and Jin. As an application, the sequence consisting of these coefficients on polymatroids is proven to be unimodal, while the unimodality of the whole coefficients of matroids was obtained in 2018 by Adiprasito, Huh and Katz using Hodge theory.

math.CO↗

arXiv2025

Coefficients of univariate Tutte polynomials with one variable fixed

It is well known that the 2-variable Tutte polynomial of a graph $G$ includes chromatic polynomial and flow polynomial of $G$, i.e. the cases of $y=0$ and $x=0$. In 2013, K\'{a}lm\'{a}n introduced the interior and exterior polynomials which generalized the cases of $y=1$ and $x=1$ of Tutte polynomials of graphs to hypergraphs, and further polymatroids. There have been some results on coefficients of these polynomials, which motivate us to study uniformly the coefficients of $T_M(x,t)$ and $T_M(t,y)$, where $T_M(x,y)$ denotes the Tutte polynomial of a matroid $M$ and $t$ is a fixed real number. In this paper, we introduce two mutually dual parameters $f_k(M)$ and $g_k(M)$ ($g_1(M)$ is the girth of $M$) for any nonnegative integer $k$, and obtain the following results: (1) Formulas for coefficients of the higher-degree terms (related to $g_2(M)$ and $f_2(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ in terms of circuits and hyperplanes of $M$; (2) when $0\leq t \leq 1$, coefficients of the more higher-degree terms (related to $g_1(M)$ and $f_1(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ are further simplified and characterized; (3) As applications, some known results in the cases $t=0$ and $t=1$ are derived and generalized, and the unimodality of these coefficients in (1) are proved when $t\leq 1$.

math.CO↗

arXiv2024

Tight Toughness and Isolated Toughness for $\{K_2,C_n\}$-factor critical avoidable graph

A spannning subgraph $F$ of $G$ is a $\{K_2,C_n\}$-factor if each component of $F$ is either $K_{2}$ or $C_{n}$. A graph $G$ is called a $(\{K_2,C_n\},n)$-factor critical avoidable graph if $G-X-e$ has a $\{K_2,C_n\}$-factor for any $S\subseteq V(G)$ with $|X|=n$ and $e\in E(G-X)$. In this paper, we first obtain a sufficient condition with regard to isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{n}\}$-factor critical avoidable. In addition, we give a sufficient condition with regard to tight toughness and isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{2i+1}|i \geqslant 2\}$-factor critical avoidable respectively.

math.CO↗

arXiv2024

A direct proof of well-definedness for the polymatroid Tutte polynomial

For a polymatroid $P$ over $[n]$, Bernardi, K\'{a}lm\'{a}n and Postnikov [\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $\mathscr{T}_{P}$ is equivalent to another polynomial that only depends on $P$. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.

math.CO↗

arXiv2024

On the connected coalition number

For a graph $G=(V,E)$, a pair of vertex disjoint sets $A_{1}$ and $A_{2}$ form a connected coalition of $G$, if $A_{1}\cup A_{2}$ is a connected dominating set, but neither $A_{1}$ nor $A_{2}$ is a connected dominating set. A connected coalition partition of $G$ is a partition $Φ$ of $V(G)$ such that each set in $Φ$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a connected coalition of $G$ with another set in $Φ$. The connected coalition number of $G$, denoted by $CC(G)$, is the largest possible size of a connected coalition partition of $G$. In this paper, we characterize graphs that satisfy $CC(G)=2$. Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.

math.CO↗

arXiv2023

A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial

The Tutte polynomial is a fundamental invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi, K\'{a}lm\'{a}n, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first obtain a deletion-contraction formula for $\mathscr{T}_{P}(x,y)$. Then we prove two natural properties of coefficientwise monotonicity, one for containment and one for minors, both for the interior polynomial $x^{n}\mathscr{T}_{P}(x^{-1},1)$ and the exterior polynomial $y^{n}\mathscr{T}_{P}(1,y^{-1})$, where $P$ is a polymatroid over $[n]$. We show by an example that these monotonicity properties do not extend to $\mathscr{T}_{P}(x,y)$. Using deletion-contraction, we obtain formulas for the coefficients of terms of degree $n-1$ in $\mathscr{T}_{P}(x,y)$. Finally, we characterize hypergraphs $\mathcal{H}=(V,E)$ such that the coefficient of $y^{k}$ in the exterior polynomial of the associated polymatroid $P_{\mathcal{H}}$ attains its maximal value $\binom{|V|+k-2}{k}$ for all $k$ up to some bound.

math.CO↗

arXiv2022

Extremal trees with respect to spectral radius of restrictedly weighted adjacency matrices

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle f(d_{i}, d_{j})$ if the vertices $v_{i}$ and $v_{j}$ are adjacent and 0 otherwise. Li and Wang \cite{U9} tried to unify methods to study spectral radius of weighted adjacency matrices of graphs weighted by various topological indices. If $\displaystyle f'_{x}(x, y)\geq0$ and $\displaystyle f''_{x}(x, y)\geq0$, then $\displaystyle f(x, y)$ is said to be increasing and convex in variable $x$, respectively. They obtained the tree with the largest spectral radius of $A_{f}(G)$ is a star or a double star when $f(x, y)$ is increasing and convex in variable $x$. In this paper, we add the following restriction: $f(x_{1},y_{1})\geq f(x_{2},y_{2})$ if $x_{1}+y_{1}=x_{2}+y_{2}$ and $\mid x_{1}-y_{1}\mid>\mid x_{2}-y_{2}\mid$ and call $A_f(G)$ the restrictedly weighted adjacency matrix of $G$. The restrictedly weighted adjacency matrix contains weighted adjacency matrices weighted by first Zagreb index, first hyper-Zagreb index, general sum-connectivity index, forgotten index, Somber index, $p$-Sombor index and so on. We obtain the extremal trees with the smallest and the largest spectral radius of $A_{f}(G)$. Our results push ahead Li and Wang's research on unified approaches.

math.CO↗

arXiv2022

On the polymatroid Tutte polynomial

The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial $\mathcal{J}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first prove that $\mathcal{J}_{P}(x,t)$ and $\mathcal{J}_{P}(t,y)$ are interpolating for any fixed real number $t\geq 1$, and then we study the coefficients of high-order terms in $\mathcal{J}_{P}(x,1)$ and $\mathcal{J}_{P}(1,y)$. These results generalize results on interior and exterior polynomials of hypergraphs.

math.CO↗

arXiv2022

Tight toughness, isolated toughness and binding number bounds for the $\{K_2,C_n\}$-factors

The $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{K_2,C_{2i+1}| i\geq 2 \}$-factor for any graph is obtained, which answers a problem due to Gao and Wang (J. Oper. Res. Soc. China (2021), https://doi.org/10.1007/s40305-021-00357-6).

math.CO↗

arXiv2022

A direct and elementary proof of the well-definedness of the interior and exterior polynomials of hypergraphs

T. Kálmán (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and $(1,1/y)$ to hypergraphs. The two polynomials are defined under a fixed ordering of hyperedges, and are proved to be independent of the ordering using techniques of polytopes. In this paper, similar to the Tutte's original proof we provide a direct and elementary proof for the well-definedness of the interior and exterior polynomials of hypergraphs.

math.CO↗

arXiv2022

On coefficients of the interior and exterior polynomials

The interior polynomial and the exterior polynomial are generalizations of valuations on $(1/ξ,1)$ and $(1,1/η)$ of the Tutte polynomial $T_G(x,y)$ of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let $G=(V\cup E, \varepsilon)$ be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in $V$ or $E$. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a `dual' result on the coefficient of the linear term of the exterior polynomial: if $G-e$ is connected for each $e\in E$, then the coefficient of the linear term of the exterior polynomial is $|V|-1$. Interior and exterior polynomials for some families of bipartite graphs are computed.

math.CO↗

arXiv2018

Anti-$k$-labeling of graphs

It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-$k$-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference. Let $k\geq1$ be an integer and $ψ$ is a labeling function (anti-$k$-labeling) from $V(G)$ to $\{1,2,\cdots,k\}$ for a graph $G$. A {\em no-hole anti-$k$-labeling} is an anti-$k$-labeling using all labels between 1 and $k$. We define $w_ψ(e)=|ψ(u)-ψ(v)|$ for an edge $e=uv$ and $w_ψ(G)=\min\{w_ψ(e):e\in E(G)\}$ for an anti-$k$-labeling $ψ$ of the graph $G$. {\em The anti-$k$-labeling number} of a graph $G$, $mc_k(G)$ is $\max\{w_ψ(G): ψ\}$. In this paper, we first show that $mc_k(G)=\lfloor \frac{k-1}{χ-1}\rfloor$, and the problem that determines $mc_k(G)$ of graphs is NP-hard. We mainly obtain the lower bounds on no-hole anti-$n$-labeling number for trees, grids and $n$-cubes.

math.CO↗

arXiv2018

Minimum degree and size conditions for the proper connection number of graphs

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $k\geq3$. If $|E(G)|\geq \binom{n-m-(k+1-m)(δ+1)}{2} +(k+1-m)\binom{δ+1}{2}+k+2$, then $pc(G)\leq k$, where $m$ takes the value $t$ if $δ=1$ and $\lfloor \frac{k}{δ-1} \rfloor$ if $δ\geq2$. Furthermore, if $k=2$ and $δ=2$, %(i.e., $|E(G)|\geq \binom{n-5}{2} +7$) $pc(G)\leq 2$, except $G\in \{G_{1}, G_{n}\}$ ($n\geq8$), where $G_{1}=K_{1}\vee 3K_{2}$ and $G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_{1}\vee (2K_{2}$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}\vee (2K_{2}$) being joined. If $k=2$, $δ\geq 3$, we conjecture $pc(G)\leq 2$, where $m$ takes the value $1$ if $δ=3$ and $0$ if $δ\geq4$ in the assumption.

math.CO↗

arXiv2018

Embedding 5-planar graphs in three pages

A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the \emph{page number}. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectured that its page number can be made arbitrarily large [JCT, 1979, 320-331]. Heath [FOCS84] found that planar graphs admit a seven-page book embedding. Later, Yannakakis proved that four pages are necessary and sufficient for planar graphs in [STOC86]. Recently, Bekos et al. [STACS14] described an $O(n^{2})$ time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an $O(n^{2})$ time algorithm.

math.CO↗