arXiv subjects

Rong Luo

Publications and source records attributed to Rong Luo.

At least 19 recordsLinked to original sources

arXiv2026

Reduction Operations and Structural Characterizations of $S^1$-Flows in Graphs

While every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow, the converse does not hold in general as shown by Thomassen (2014). In this paper, we develop reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and wheel contractions. A key tool is the two-terminal $S^1$-preflow, which enables us to prove that if a $2$-connected graph contains an odd wheel as a proper subgraph and contracting the wheel yields a graph with a nowhere-zero $3$-flow, then the original graph admits an $S^1$-flow. As applications, we completely characterize $S^1$-flows in two graph classes: a triangularly connected graph admits an $S^1$-flow if and only if it is not an odd wheel; and a graph containing a spanning triangle-tree admits an $S^1$-flow if and only if it is not an odd crystal.

math.CO↗

arXiv2026

On the Spectra of Chromatic Number and Chromatic Index of Cyclic Covers

For a fixed integer $\ell \ge 2$, we study what values of chromatic index and chromatic number can be attained by some $\ell$-fold cyclic cover of a loopless multigraph. For edge-coloring, we first investigate the density, a fundamental lower bound for the chromatic index, and show that the density of every $\ell$-fold cyclic cover of a graph $G$ is at most that of $G$. We further prove that if $\ell$ is even, then the spectrum of chromatic indices over all $\ell$-fold cyclic covers of $G$ contains every integer between $\Delta(G)$ and $\chi'(G)$. When $\ell$ is odd, the chromatic-index spectrum need not be complete in general; for edge-chromatic critical graphs, we determine exactly which values are attainable. For vertex-coloring, we prove that if $\chi(G)\ge 3$, then the spectrum of chromatic numbers over all $\ell$-fold cyclic covers of $G$ contains every integer between $3$ and $\chi(G)$. Moreover, this spectrum contains $2$ if and only if $G$ is bipartite or $\ell$ is even.

math.CO↗

arXiv2026

The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$

The Alon--Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number $\chi_{\ell}(G)$. We study the Alon--Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that $\chi_{\ell}(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.

math.CO↗

arXiv2026

High-Dimensional $p$-Normed Flows

We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$ to be the infimum over all real numbers $r$ for which $G$ admits a $d$-dimensional $p$-normed nowhere-zero $r$-flow. For any bridgeless graph $G$ and any $p\ge 1$, we establish general upper bounds, including $\phi_{2,p}(G) \le 3$, $\phi_{3,p}(G) \le 1+\sqrt{2}$, and tight bounds for graphs admitting a $4$-NZF. For graphs with oriented $(k+1)$-cycle $2l$-covers, we show that $\phi_{k,p}(G) = 2$, which implies $\phi_{2,p}(G) = 2$ for graphs admitting a nowhere-zero $3$-flow and $\phi_{3,p}(G) = 2$ for those admitting a nowhere-zero $4$-flow. These results extend classical flow theory to arbitrary norms, provide supporting evidences for Tutte's $5$-flow Conjecture and Jain's $S^2$-Flow Conjecture, and connect combinatorial flows with geometric and topological perspectives.

math.CO↗

arXiv2025

Squares of subcubic planar graphs without cycles of length 4-8 are 6-choosable

The {\em square} of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and an edge between any two vertices at distance at most $2$ in $G$. Wegner (1977) conjectured that for a planar graph $G$, $\chi(G^2) \leq 7$ if $\Delta(G) = 3$, $\chi(G^2) \leq \Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$, and $\chi(G^2) \leq \lfloor 3\Delta(G)/2 \rfloor$ if $\Delta(G) \geq 8$, and Thomassen (2018) confirmed the conjecture for $\Delta(G) = 3$. Dvo\v{r}\'{a}k et al. (2008) and Feder et al. (2021) further conjectured that $\chi(G^2) \leq 6$ for cubic bipartite planar graphs. A natural question is whether this bound also holds for the list-chromatic number, i.e., whether $\chi_{\ell}(G^2) \leq 6$ for such graphs. More generally, it is of interest to determine sufficient conditions ensuring $\chi_{\ell}(G^2) \leq 6$ for subcubic planar graphs. In this paper, we prove that $\chi_{\ell}(G^2) \leq 6$ for subcubic planar graphs containing no $k$-cycles for $4 \leq k \leq 8$, improving a result of Cranston and Kim (2008).

math.CO↗

arXiv2025

PORTool: Importance-Aware Policy Optimization with Rewarded Tree for Multi-Tool-Integrated Reasoning

Multi-tool-integrated reasoning enables LLM-empowered tool-use agents to solve complex tasks by interleaving natural-language reasoning with calls to external tools. However, training such agents from outcome-only rewards suffers from credit-assignment ambiguity, obscuring which intermediate tool-use decisions drive success or failure. In this paper, we propose PORTool, an importance-aware policy-optimization algorithm that reinforces agents' tool-use competence from outcome-level supervision while assigning reward at the step level. Specifically, PORTool generates a rewarded rollout tree in which trajectories share prefixes before branching, enabling direct comparisons among alternative tool-use decisions within the same context. It then estimates each step's importance by a correctness-dominant signal, i.e., whether descendants of that step can ultimately produce a correct final answer, plus an auxiliary term indicating whether the step's tool calls satisfy formatting constraints and execute successfully. Using these step-wise importance estimates, PORTool updates the policy to generate efficient tool-call steps, guided by both local comparisons within each branching decision and the overall quality of entire trajectories. Experiments show that PORTool improves final-answer accuracy while reducing tool-call steps compared with state-of-the-art policy-optimization baselines, and ablation studies confirm the robustness of the proposed step-wise importance estimates.

cs.CL↗

arXiv2025

Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes

In this paper, we establish the conditions for some finite abelian groups and the family all the $k$-sets in each of them summing up to an element $x$ to form $t$-designs. We fully characterize the sufficient and necessary conditions for the incidence structures to form $1$-designs in finite abelian $p$-groups, generalizing existing results on vector spaces over finite fields. For finite abelian groups of exponent $pq$, we also propose sufficient and necessary conditions for the incidence structures to form a $1$-designs. Furthermore, some interesting observations of the general case when the group is cyclic or non-cyclic are presented and the relations between $(t-1)$-designs and $t$-designs from subset sums are established. As an application, we demonstrate the correspondence between $t$-designs from the minimum-weight codewords in elliptic curve codes and subset-sum designs in their groups of rational points. By such a correspondence, elliptic curve codes supporting designs can be simply derived from subset sums in finite abelian groups that supporting designs.

math.CO↗

arXiv2024

An 8-flow theorem for signed graphs

We prove that a signed graph admits a nowhere-zero $8$-flow provided that it is flow-admissible and the underlying graph admits a nowhere-zero $4$-flow. When combined with the 4-color theorem, this implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $8$-flow. Our result improves and generalizes previous results of Li et al. (European J. Combin. 108 (2023), 103627), which state that every flow-admissible signed $3$-edge-colorable cubic graph admits a nowhere-zero $10$-flow, and that every flow-admissible signed hamiltonian graph admits a nowhere-zero $8$-flow.

math.CO↗

arXiv2023

Signed circuit $6$-covers of signed $K_4$-minor-free graphs

Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit $k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is the well-know circuit double cover conjecture. For signed circuit $k$-cover of signed graphs, it is known that for every integer $k\leq 5$, there are infinitely many coverable signed graphs without signed circuit $k$-cover and there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't admit a signed circuit $1$-cover. Fan conjectured that every coverable signed graph has a signed circuit $6$-cover. This conjecture was verified only for signed eulerian graphs and for signed graphs whose bridgeless-blocks are eulerian. In this paper, we prove that this conjecture holds for signed $K_4$-minor-free graphs. The $6$-cover is best possible for signed $K_4$-minor-free graphs.

math.CO↗

arXiv2023

Integer flows on triangularly connected signed graphs

A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in $G$ such that for any $i, j$ with $1\leq i < j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j > i+1$. A connected graph $G$ is triangularly connected if for any two nonparallel edges $e$ and $e'$ there is a triangle-path $T_1T_2\cdots T_m$ such that $e\in E(T_1)$ and $e'\in E(T_m)$. For ordinary graphs, Fan {\it et al.}~(J. Combin. Theory Ser. B 98 (2008) 1325-1336) characterize all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result include integer flow of some families of ordinary graphs, such as, locally connected graphs due to Lai (J. Graph Theory 42 (2003) 211-219) and some types of products of graphs due to Imrich et al.(J. Graph Theory 64 (2010) 267-276). In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that every flow-admissible triangularly connected signed graph admits a nowhere-zero $4$-flow if and only if it is not the wheel $W_5$ associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere-zero $4$-flow but not $3$-flow.

math.CO↗

arXiv2023

New sets of Non-Orthogonal Spreading Sequences With Low Correlation and Low PAPR Using Extended Boolean Functions

Extended Boolean functions (EBFs) are one of the most important tools in cryptography and spreading sequence design in communication systems. In this paper, we use EBFs to design new sets of spreading sequences for non-orthogonal multiple access (NOMA), which is an emerging technique capable of supporting massive machine-type communications (mMTC) in 5G and beyond. In this work, first $p$-ary complementary sequences are constructed using EBFs and then, these sequences are used to design new sets of non-orthogonal spreading sequence sets having very low coherence and peak to average power ratio (PAPR). The proposed spreading sequence sets are capable of supporting a large number of active devices simultaneously. In fact, for a $p$-ary spreading sequence set, we theoretically achieve an overloading factor of $2p$, where $p$ is an odd prime. Specifically, for $p=3$, we achieve an overloading factor of $6$, which cannot be achieved through the existing constructions till date.

cs.IT↗

arXiv2023

The average degree of edge chromatic critical graphs with maximum degree seven

In this paper, by developing several new adjacency lemmas about a path on $4$ or $5$ vertices, we show that the average degree of 7-critical graphs is at least 6. It implies Vizing's planar graph conjecture for planar graphs with maximum degree $7$ and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao (J. Combin. Theory Ser. B 83 (2001) 201-212 and J. Combin. Theory Ser. B 87 (2003) 254-263) and Zhang (Graphs and Combinatorics 16 (2000) 467-495).

math.CO↗

arXiv2022

Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences

Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $C_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$).

cs.IT↗

arXiv2022

Decomposition of class II graphs into two class I graphs

Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $\Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $\Delta(G)+k$ ($k\geq 1$) can be decomposed into a maximum $\Delta(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $\mu$ can be decomposed into a maximum $\Delta(G)$-edge-colorable subgraph and a subgraph with maximum degree at most $\mu$. Then we prove that every graph $G$ with chromatic index $\Delta(G)+k$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that $\Delta(H_1) = \Delta(G)$ and $\Delta(H_2) = k$, which is a variation of their conjecture.

math.CO↗

arXiv2022

Flows of 3-edge-colorable cubic signed graphs

Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-zero $10$-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.

math.CO↗

arXiv2022

Two classes of subfield codes of linear codes

Recently, subfiled codes of linear code over GF$ (q) $ with good parameters were studied, and many optimal subfield codes were obtained. In this paper, Our mainly motivation is to generlize the results of the subfield codes of hyperoval in Ding and Heng (Finite Fields Their Appl. 56, 308-331 (2019)), and generlize the results of two families of subfield codes in Xiang and Yin (Cryptogr. Commun. 13(1), 117-127 (2021)) to $ p $-ary where $ p $ is odd. We get the parameters and weight distribution of these subfield codes. At the same time, the parameters of their dual codes are also studied. When $ m=1 $, The dual codes of these subfield codes are almost MDS code, when $ m>1 $ and $ p $ odd, these dual codes are dimension-optimal with respect to the sphere-backing bound.

cs.IT↗

arXiv2022

MDS and AMDS symbol-pair codes constructed from repeated-root codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length $lp$. In addition, we derive all MDS and AMDS symbol-pair codes with length $3p$, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.

cs.IT↗

arXiv2022

MDS and AMDS symbol-pair codes are constructed from repeated-root codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this paper, we construct more general generator polynomials for two classes of MDS symbol-pair codes with code length $lp$. Based on repeated-root cyclic codes, we derive all MDS symbol-pair codes of length $3p$, when the degree of the generator polynomials is no more than 10. We also give two new classes of (almost maximal distance separable) AMDS symbol-pair codes with the length $lp$ or $4p$ by virtue of repeated-root cyclic codes. For length $3p$, we derive all AMDS symbol-pair codes, when the degree of the generator polynomials is less than 10. The main results are obtained by determining the solutions of certain equations over finite fields.

cs.IT↗