arXiv subjects

Rongxing Xu

Publications and source records attributed to Rongxing Xu.

23 records · Page 2Linked to original sources

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↗

arXiv2020

Multiple list colouring 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 characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed that if $G=Θ_{r,s,t}$ where $r,s,t$ have the same parity and $\min\{r,s,t\} \ge 3$, or $G=Θ_{2,2,2,2p}$ with $p \ge 2$, then $G$ is bipartite $3$-choice critical, but not $(4,2)$-choosable. On the other hand, all the other bipartite 3-choice critical graphs are $(4,2)$-choosable. This paper strengthens the result of Meng, Puleo and Zhu and shows that all the other bipartite $3$-choice critical graphs are $(4m,2m)$-choosable for every integer $m$.

math.CO↗

arXiv2019

A New Strategy in Applying the Learning Machine to Study Phase Transitions

In this Letter, we present a new strategy for applying the learning machine to study phase transitions. We train the learning machine with samples only obtained at a non-critical parameter point, aiming to establish intrinsic correlations between the learning machine and the target system. Then, we find that the accuracy of the learning machine, which is the most important performance index in conventional learning machines, is no longer a key goal of the training in our approach. Instead, relatively low accuracy of identifying unlabeled data category can help to determine the critical point with greater precision, manifesting the singularity around the critical point. It thus provides a robust tool to study the phase transition. The classical ferromagnetic and percolation phase transitions are employed as illustrative examples.

cond-mat.stat-mech↗

arXiv2017

List colouring of graphs and generalized Dyck paths

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V(G) \to N$ is a mapping. For a nonnegative integer $m$, let $f^{(m)}$ be the extension of $f$ to the graph $ G \diamondplus \overline{K_m}$ for which $f^{(m)}(v)=|V(G)|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-paintable. We study the parameter $m_c(K_n, f)$ and $m_p(K_n,f)$ for arbitrary mappings $f$. For $\vec{x}=(x_1,x_2,\ldots,x_n)$, an $\vec{x}$-dominated path ending at $(a, b)$ is a monotonic path $P$ of the $a \times b$ grid from $(0,0)$ to $(a,b)$ such that each vertex $(i,j)$ on $P$ satisfies $i \le x_{j+1}$. Let $ψ(\vec{x})$ be the number of $\vec{x}$-dominated paths ending at $(x_n,n)$. By this definition, the Catalan number $C_n$ equals $ ψ((0,1, \ldots, n-1)) $. This paper proves that if $G=K_n$ has vertices $v_1, v_2, \ldots, v_n$ and $f(v_1) \le f(v_2) \le \ldots \le f(v_n)$, then $m_c(G,f)=m_p(G,f)=ψ(\vec{x}(f))$, where $\vec{x}(f)=(x_1, x_2, \ldots, x_n)$ and $x_i=f(v_i)-i$ for $i=1, 2,\ldots, n$. Therefore, if $f(v_i)=n$, then $m_c(K_n, f)=m_p(K_n, f)$ equals the Catalan number $C_n$.

math.CO↗