arXiv subjects

Rongxing Xu

Publications and source records attributed to Rongxing Xu.

At least 19 recordsLinked to original sources

arXiv2026

On the maximum number of triangles in tripartite graphs with no $4$-cycles between any two parts

Let $G$ be a $3$-partite graph with $k$ vertices in each part such that the bipartite graph induced by any two parts contains no cycle of length four. Fischer and Matoušek [J. Combin. Theory Ser. A, 2001] asked for the maximum number of triangles in such a graph. They obtained the lower bound $(1-o(1))k^{3/2}$ and the upper bound $k^{7/4}+O(k^{3/2})$. Coulter, Matthews and Timmons [J. Combin. Theory Ser. B, 2018] later constructed such graphs using planar polynomials over finite fields and improved the lower bound to $(1-o(1))k^{5/3}$. In this note, we use a new triple of planar polynomials and further improve the lower bound to $(1-o(1))k^{17/10}$.

math.CO↗

arXiv2026

Coarse-grained kinetic scale tightens thermodynamic spectral bounds of Markov cycles

Thermodynamic bounds on the spectrum of a Markov cycle depend on the cycle affinity and maximum escape rate, but they ignore how the local transition rates are distributed around the cycle. In this work, we show that a single additional cycle-wide kinetic quantity, the geometric means of forward and backward rates, yields a strictly stronger bound for every nonuniform cycle. By comparing each winding sector of the eigenmode with a product-matched uniform cycle, we bound oscillation frequencies, tighten the admissible spectral region, and show that only uniform cycles can saturate the Uhl-Seifert boundary. Our results show that retaining a coarse-grained kinetic scale can sharpen thermodynamic spectral bounds without requiring knowledge of the full generator. Our method also provides a complex-analytic framework which unifies generator-specific kinetic information with established affinity-winding bounds.

cond-mat.stat-mech↗

arXiv2026

The strong fractional choice number of triangle-free planar graphs

Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$ such that $C(u)\cap C(v)=\emptyset$ whenever $uv\in E(G)$. We show that every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most $15/4$. This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case $m=1$ answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019].

math.CO↗

arXiv2026

Codegree Thresholds for $λ$-Choosability of Graphs

Let $λ=\{k_1,\ldots,k_q\}$ be a partition, and let $|λ|=k_1+\cdots+k_q$. A $|λ|$-list assignment $L$ of a graph $G$ is a $λ$-assignment if its color set can be partitioned into $q$ disjoint sets $X_1,\ldots,X_q$ such that $|L(v)\cap X_i|=k_i$ for every vertex $v$ and every $i\in[q]$. This notion, introduced by Zhu [J. Combin. Theory Ser. B, 2020], puts ordinary coloring and list coloring in the same framework. A theorem of Alon [Random Structures Algorithms, 2000] states that every graph with minimum degree $d$ has choice number at least $(1/2-o(1))\log_2d$. Saxton and Thomason [Invent. Math., 2015] later used the hypergraph container method to replace $1/2$ by the sharp constant $1$. It is natural to ask whether a similar phenomenon holds for every fixed partition $λ$. Minimum degree alone is not sufficient: balanced complete bipartite graphs have arbitrarily large minimum degree but are always $\{1,1\}$-choosable. We show that the appropriate replacement is the minimum $q$-codegree, defined for $|V(G)|\geq q$ by $δ_q(G)=\min\{|N_G(S)|:S\subseteq V(G),\,|S|=q\}$. More precisely, for every partition $λ$ there exists an integer $d$ such that every graph $G$ with $δ_q(G)\geq d$ is not $λ$-choosable. Let $f(λ)$ be the least such $d$. For every fixed $q$, we prove $f(λ)\leq2^{(2q+o(1))|λ|}$ as $|λ|\to\infty$, while $f(λ)\geq(q+1)^{-1}(1+1/q)^{|λ|}$ for every $λ$. For the partition $\{k,\ldots,k\}$ with $q$ equal parts, we determine the threshold asymptotically: $f(\{k,\ldots,k\})=ρ_q^{-(1+o(1))k}$ as $k\to\infty$, where $ρ_q$ is the unique $x\in(0,1)$ satisfying $x=(1-x)^q$. When $q=1$, our result implies $\operatorname{ch}(G)\geq(1-o(1))\log_2δ(G)$.

math.CO↗

arXiv2026

Degree-choosability of proper conflict-free list coloring of sparse graphs

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $ϕ$ of $G$ such that for every vertex $v \in V(G)$, $ϕ(v) \in L(v)$ and some appears precisely once in the neighborhood of $v$. We say that $G$ is proper conflict-free $f$-choosable if for every $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. If $G$ is proper conflict-free $f$-choosable and there is a constant $k$ such that $f(v)= d_G(v)+k$ for every vertex $v$ of $G$, then we say $G$ is proper conflict-free $({\rm degree}+k)$-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than $\frac{10}{3}$ is proper conflict-free $({\rm degree}+3)$-choosable, and that every graph with the maximum average degree less than $\frac{18}{7}$ is proper conflict-free $({\rm degree}+2)$-choosable. As a result, every planar graph with girth at least $5$ is proper conflict-free $({\rm degree}+3)$-choosable, and every planar graph with girth at least $9$ is proper conflict-free $({\rm degree}+2)$-choosable.

math.CO↗

arXiv2025

Partitioning triangle-free planar graphs into a forest and a linear forest

Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another proof of a result of Borodin and Glebov, showing this result for planar graphs of girth~5. Subsequently, Dross, Montassier, and Pinlou raised the same question and proved that every triangle-free planar graph can be partitioned into a forest and another forest of maximum degree~5. More recently, Feghali and Šámal improved this bound on the maximum degree to~3. In this note, we further improve the result by showing that every triangle-free planar graph can be partitioned into a forest and a linear forest, that is, a forest of maximum degree~2.

math.CO↗

arXiv2025

Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy

A proper coloring $ϕ$ of $G$ is called a proper conflict-free coloring of $G$ if for every non-isolated vertex $v$ of $G$, there is a color $c$ such that $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ for every vertex $v\in V(G)$. In this note, we first remark if a graph $G$ is $d$-degenerate, then $G$ is proper conflict-free $({\rm degree}+d+1)$-choosable. Furthermore, when $d=1$, we can reduce the number of colors by showing that every tree is proper conflict-free $({\rm degree}+1)$-choosable. This motivates us to state a question.

math.CO↗

arXiv2025

Proper conflict-free degree-choosability of outerplanar graphs

A proper coloring $ϕ$ of $G$ is called a proper conflict-free coloring of $G$ if for every non-isolated vertex $v$ of $G$, there is a color $c$ such that $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy to degree-choosability of graphs, the authors recently, in a previous paper, introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ for every vertex $v\in V(G)$. In this paper, we show that every connected outerplanar graph other than the $5$-cycle is proper conflict-free $({\rm degree}+2)$-choosable. This bound is tight in the sense that there are infinitely many connected outerplanar graphs that are not proper conflict-free $({\rm degree}+1)$-choosable. We conclude the paper with two questions for further work.

math.CO↗

arXiv2025

Proof of a conjecture of Voss on bridges of longest cycles

Bridges are a classical concept in structural graph theory and play a fundamental role in the study of cycles. A conjecture of Voss from 1991 asserts that if disjoint bridges $B_1, B_2, \ldots, B_k$ of a longest cycle $L$ in a $2$-connected graph overlap in a tree-like manner (i.e., induce a tree in the {\it overlap graph} of $L$), then the total {\it length} of these bridges is at most half the length of $L$. Voss established this for $k \leq 3$ and used it as a key tool in his 1991 monograph on cycles and bridges. In this paper, we confirm the conjecture in full via a reduction to a cycle covering problem.

math.CO↗

arXiv2025

Results on proper conflict-free list coloring of graphs

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $ϕ$ of $G$ such that $ϕ(v) \in L(v)$ for every vertex $v\in V(G)$ and $v$ has a color that appears precisely once at its neighborhood for every non-isolated vertex $v\in V(G)$. We say that $G$ is proper conflict-free $f$-choosable if for any $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. For a non-negative integer $k$, we say that $G$ is \emph{proper conflict-free $({\rm degree}+k)$-choosable} if $G$ is proper conflict-free $f$-choosable where $f$ is a mapping with $f(v)= d_G(v)+k$ for every vertex $v\in V(G)$. Motivated by degree-choosability of graphs, we investigate the proper conflict-free $({\rm degree}+k)$-choosability of graphs, especially for cases $k=1,2,3$. As the 5-cycle is not proper conflict-free $({\rm degree}+2)$-choosable and it is the only such graph we know, it is possible that every connected graph other than the 5-cycle is proper conflict-free $({\rm degree}+2)$-choosable and thus every graph is proper conflict-free $({\rm degree}+3)$-choosable. To support these, we show that every connected graph with maximum degree at most 3 distinct from the 5-cycle is proper conflict-free $(\text{degree}+2)$-choosable, and that $S(G)$ is proper conflict-free $(\text{degree}+2)$-choosable for every graph $G$, where $S(G)$ is a graph obtained from $G$ by subdividing each edge once. Furthermore, by adapting the technique of DP-colorings, we prove that every graph with maximum degree at most $4$ is proper conflict-free $({\rm degree}+3)$-choosable.

math.CO↗

arXiv2023

On Two problems of defective choosability

Given positive integers $p \ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\geq k$ for each $v \in V(G)$ and $|\bigcup_{v\in V(G)}L(v)| \leq p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\infty)$-choosable means $k$-choosable and $(k,d,+\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any positive integers $\ell \geq k \geq 3$, and non-negative integer $d$, there are $(k,d, \ell)$-choosable graphs that are not $(k,d , \ell+1)$-choosable. These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27, 4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353], respectively. Our construction of $(k,d, \ell)$-choosable but not $(k,d , \ell+1)$-choosable graphs generalizes the construction of Král' and Sgall in [J. Graph Theory 49, 3(2005), 177-186] for the case $d=0$.

math.CO↗

arXiv2022

Decomposition of triangle-free planar graphs

A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E(G)$. In this paper, we prove that every triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph and a matching. Consequently, every triangle-free planar graph $G$ has a matching $M$ such that $G-M$ is online 3-DP-colorable. This strengthens an earlier result in [R. Škrekovski, {\em A Grötzsch-Type Theorem for List Colourings with Impropriety One}, Combin. Prob. Comput. 8 (1999), 493-507] that every triangle-free planar graph is $1$-defective $3$-choosable.

math.CO↗

arXiv2021

Reinforcement Learning Approach to Shortcuts between Thermodynamic States with Extra Constraints

We propose a systematic method based on reinforcement learning (RL) techniques to find the optimal path that can minimize the total entropy production between two equilibrium states of open systems at the same temperature in a given fixed time period. Benefited from the generalization of the deep RL techniques, our method can provide a powerful tool to address this problem in quantum systems even with two-dimensional continuous controllable parameters. We successfully apply our method on the classical and quantum two-level systems.

quant-ph↗

arXiv2021

Extended Double Covers and Homomorphism Bounds of Signed Graphs

A \emph{signed graph} $(G, σ)$ is a graph $G$ together with an assignment $σ:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connection through the notion of Extended Double Covers of signed graphs, which was recently introduced by Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B, π)$ is planar-complete if any planar signed graph $(G, σ)$ which verifies the conditions of a basic no-homomorphism lemma with respect to $(B,π)$ admits a homomorphism to $(B, π)$. Our conjecture then is that: if $(B, π)$ is a connected signed graph with no positive odd closed walk which is planar-complete, then its Extended Double Cover ${\rm EDC}(B,π)$ is also planar-complete. We observe that this conjecture largely extends the Four-Color Theorem and is strongly connected to a number of conjectures in extension of this famous theorem. A given (signed) graph $(B,π)$ \emph{bounds} a class of (signed) graphs if every (signed) graph in the class admits a homomorphism to $(B,π)$. In this work, and in support of our conjecture, we prove it for the subclass of signed $K_4$-minor free graphs. Inspired by this development, we then investigate the problem of finding optimal homomorphism bounds for subclasses of signed $K_4$-minor-free graphs with restrictions on their girth and we present nearly optimal solutions. Our work furthermore leads to the development of weighted signed graphs.

math.CO↗

arXiv2021

The strong fractional choice number and the strong fractional paint number of graphs

This paper studies the strong fractional choice number $ch^s_f(G)$ and the strong fractional paint number $χ^s_{f,P}(G)$ of a graph $G$. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any positive integers $p,q$ satisfying $2 \le \frac{2p}{2q+1} \leq \lfloor\frac{p}{q}\rfloor$, there exists a graph $G$ with $ch^s_f(G) = χ^s_{f,P}(G) = \frac{p}{q}$. The relationship between $χ^s_{f,P}(G)$ and $ch^s_f(G)$ is explored. We prove that the gap $χ^s_{f,P}(G)-ch^s_f(G)$ can be arbitrarily large. The strong fractional choice number of a family $\mathcal{G}$ of graphs is the supremum of the strong fractional choice number of graphs in $\mathcal{G}$. Let $\mathcal{P}$ denote the class of planar graphs and $\mathcal{P}_{k_1,\ldots, k_q}$ denote the class of planar graphs without $k_i$-cycles for $i=1,\ldots, q$. We prove that $3 + \frac{1}{2} \leq ch^s_f(\mathcal{P}_{ 4}) \leq 4$, $ch^s_f(\mathcal{P}_{ k})=4$ for $k \in \{5,6\}$, $3 +\frac{1}{12} \leq ch^s_f(\mathcal{P}_{ 4,5}) \leq 4$ and $ch^s_f(\mathcal{P}) \ge 4+\frac 13$. The last result improves the lower bound $4+\frac 29$ in [X. Zhu, multiple list colouring of planar graphs, Journal of Combin. Th. Ser. B,122(2017),794-799].

math.CO↗

arXiv2021

A Numerical Method to Find the Optimal Thermodynamic Cycle in Microscopic Heat Engine

Heat engines are fundamental physical objects to develop nonequilibrium thermodynamics. The thermodynamic performance of the heat engine is determined by the choice of cycle and time-dependence of parameters. Here, we propose a systematic numerical method to find a heat engine cycle to optimize some target functions. We apply the method to heat engines with slowly varying parameters and show that the method works well. Our numerical method is based on the genetic algorithm which is widely applied to various optimization problems.

cond-mat.stat-mech↗

arXiv2021

Mapping sparse signed graphs to $(K_{2k}, M)$

A homomorphism of a signed graph $(G, σ)$ to $(H, π)$ is a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ such that adjacencies, incidences and the product of signs of closed walks are preserved. Motivated by reformulations of the $k$-coloring problem in this language, and specially in connection with results on $3$-coloring of planar graphs, such as Grötzsch's theorem, in this work we consider bounds on maximum average degree which are sufficient for mapping to the signed graph $(K_{2k}, σ_m)$ ($k\geq 3$) where $σ_m$ assigns to edges of a perfect matching the negative sign. For $k=3$, we show that the maximum average degree strictly less than $\frac{14}{5}$ is sufficient and that this bound is tight. For all values of $k\geq 4$, we find the best maximum average degree bound to be 3. While the homomorphisms of signed graphs is relatively new subject, through the connection with the homomorphisms of $2$-edge-colored graphs, which are largely studied, some earlier bounds are already given. In particular, it is implied from Theorem 2.5 of "Borodin, O. V., Kim, S.-J., Kostochka, A. V., and West, D. B., Homomorphisms from sparse graphs with large girth. J. Combin. Theory Ser. B (2004)" that if $G$ is a graph of girth at least 7 and maximum average degree $\frac{28}{11}$, then for any signature $σ$ the signed graph $(G,σ)$ maps to $(K_6, σ_m)$. We discuss applications of our work to signed planar graphs and, among others, we propose questions similar to Steinberg's conjecture for the class of signed bipartite planar graphs.

math.CO↗

arXiv2020

The strong fractional choice number of $3$-choice critical graphs

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A graph $G$ is strongly fractional $r$-choosable if $G$ is $(a,b)$-choosable for all positive integers $a,b$ for which $a/b \ge r$. The strong fractional choice number of $G$ is $ch_f^s(G) = \inf \{r: G $ is strongly fractional $r$-choosable$\}$. This paper determines the strong fractional choice number of all $3$-choice critical graphs.

math.CO↗