Hamed Hatami
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $Δ$ and with no isolated edges has an \avd-coloring with at most $Δ+300$ colors, provided that $Δ>10^{20}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any linear extension of $Q_L$ generates a distinguishing 2-coloring of $L$. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.
Upper bounds for the list-distinguishing chromatic number
本文针对简单有限连通图，证明了Brooks定理在列表可区分色数上的类比，并利用着色数和列表色数给出两个上界，同时确定了书图等多种图族的列表可区分色数。
The distinguishing chromatic number of a graph $G$ is the smallest number of colors needed to properly color the vertices of $G$ so that the trivial automorphism is the only symmetry of $G$ that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4.
The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. The distinguishing chromatic index $χ'_D (G)$ of a graph $G$ is the least number $d$ such that $G$ has a proper edge labeling with $d$ labels that is preserved only by the identity automorphism of $G$. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.
For a tree $T$, consider its smallest subtree $T^{\circ}$ containing all vertices of degree at least $3$. Then the remaining edges of $T$ lie on disjoint paths each with one endpoint on $T^{\circ}$. We show that the chromatic symmetric function of $T$ determines the size of $T^{\circ}$, and the multiset of the lengths of these incident paths. In particular, this generalizes a proof of Martin, Morin, and Wagner that the chromatic symmetric function distinguishes spiders.
The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $Γ(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-Ω(\log n)}$ we have, $χ_D(Γ)\leχ(Γ) + 1$.
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $χ_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $Δ(G)$, then $χ_{D}(G)\leq Δ(G)+1$. We also obtain an upper bound for $χ_{D}(G)$ where $G$ is a graph with at most one cycle. Finally, we state a relationshi…
On the distinguishing chromatic number in hereditary graph classes
本文研究遗传图类中的区分色数，证明在禁止某些小图作为诱导子图时，区分色数的上界可显著降低。
The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.
A graph $G$ is Ramsey for a graph $H$ if every colouring of the edges of $G$ in two colours contains a monochromatic copy of $H$. Two graphs $H_1$ and $H_2$ are Ramsey equivalent if any graph $G$ is Ramsey for $H_1$ if and only if it is Ramsey for $H_2$. A graph parameter $s$ is Ramsey distinguishing if $s(H_1)\neq s(H_2)$ implies that $H_1$ and $H_2$ are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by $χ'_a(G)$. In this paper, we prove that $χ_a'(G)$ <= 5($Δ+2$)/2 for any graph G having maximum degree $Δ$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $χ_a'(G)$ <= 3$Δ$ for any graph G without is…
Distinguishing chromatic number of middle and subdivision graphs
本文研究了中间图和细分图的区分色数，通过应用已有结果确定中间图的区分色数，并证明细分图的区分数与总区分数相关，给出了细分图的区分色数上界。
A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges receive different colours and every vertex and incident edge receive different colours. This paper considers a strengthening of this condition and examines the minimum number of colours required for a total colouring with the additional property that for any adjacent vertices $u$ and $v$, the sets of colours incident to $u$ is different from the set of colours i…
Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In fact, we prove that chi D(G) \leq chi L(G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with the property that chi D(G)= chi L(G) = 3.
Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of $T$ by their numbers of edges and leaves. We prove that $S_T = <Φ,X_T>$, where $<\cdot,\cdot>$ is the Hall inner product on symmetric functions and $Φ$ is a certain symmetric function that does not depend on $T$. Thus the chromatic symmetric function is a stronge…
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factor…
Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. Finck characterized the class of graphs that satisfy equality in this bound. In this paper, we provide a new characterization of this class of graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of graphs in this class. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromati…
An adjacent vertex distinguishing total $k$-coloring $f$ of a graph $G$ is a proper total $k$-coloring of $G$ such that no pair of adjacent vertices has the same color sets, where the color set at a vertex $v$, $C^G_f(v)$, is $\{f(v)\} \cup \{f(vu)|u \in V (G), vu \in E(G)\}$. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph $G$ has adjacent vertex distinguishing total $(Δ(G)+3)$-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and $l$-coronas of graphs, not relating the results to particular graph classes.
The List-distinguishing chromatic number of graphs containing only small complete bigraphs
本文受Catlin组合技术启发，为仅含小完全二部图导出子图的连通图类建立了可区别色数与列表可区别色数的改进上界，并应用Menger定理展示了结果在基于Paley图、Dihedral群Cayley图等构造的图中的应用。
Adjacent vertex distinguishing total chromatic number of graph products
本文针对笛卡尔乘积、字典序乘积等多类图乘积，验证了邻点可区别全染色猜想（AVD-TCC）的正确性，拓展了该图论猜想的适用范围。
An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $Δ+CΔ^{\frac{1}{2}}(\logΔ)^4$, where $Δ$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.
In this article, we consider a proper total coloring distinguishes adjacent vertices by sums, if every two adjacent vertices have different total sum of colors of the edges incident to the vertex and the color of the vertex. Pilsniak and Wozniak \cite{PW} first introduced this coloring and made a conjecture that the minimal number of colors need to have a proper total coloring distinguishes adjacent vertices by sums is less than or equal to the maximum degree plus $3$. We study proper total colorings distinguishing adjacent vertices by sums of some graphs and their products. We find that these…
We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $χ_D(G)$, and the distinguishing chromatic index $χ'_D(G)$, for vertex and edge colourings, respectively. We show that $χ_D(G)\leq 2Δ(G)-1$ for graphs with a finite maximum degree $…
Vertex-distinguishing chromatic index of digraphs
该研究聚焦有向图顶点可区别色指数问题，证实了Li等人2016年提出的“至多含一个源点和一个汇点的有向图的顶点可区别色指数等于由出度、入度序列确定的自然下界”的猜想。
In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishes nonisomorphic trees with the same number of vertices, is still open in general. For special trees it has already been shown. In 2008, Martin, Morin and Wagner proved it for spiders and some caterpillars. We decompose a tree by separating the leafs and their neighbors and do the same to the remai…
The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K_2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matousek and Ziegler who recognized a hierarchy of the different topological e…
An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting problem has a close relationship with special factors of graphs. In this paper, we obtain several results on the existence of factors with the pre-specified degrees, which generalizes earlier results in [2, 3]. Using these results, we investigate edge-weighting problem. In particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weight…
The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.
We establish closed-form expansions for the number of colorings of a path or cycle on n vertices with colors from 1,...,x such that adjacent vertices are colored differently or with colors from y+1,...x.
In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.
Some Results on Local Distance Antimagic Chromatic Number of Graphs
本文针对不含孤立顶点的图，研究图的联图及图与空图的字典积的局部距离反魔法色数，推进了该图论参数的相关研究。
Quasisymmetric functions distinguishing trees
本文提出色拟对称函数可区分有向树的猜想，并通过推广 Hasebe 与 Tsujie 关于有根树的 $P$ 分拆计数函数结果，为该猜想提供支持。
A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex. The smallest value $k$ for which $G$ admits such coloring is denoted by $χ'_a(G)$. We prove that $χ'_a(G) = 2R + 1$ for most circulant graphs $C_n([1, R])$.
Distinguishing adjacent vertices by ordering edges
本文研究通过边排序区分相邻顶点的问题，证明在特定条件下（如连通非正则图或第二类图）存在全局边序使序列区分相邻顶点，并对高度正则图给出概率性结论。
A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $Δ$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $Δ+ 300$ colours, provided $Δ$ is large enough. We show that this bound can be reduced to $Δ+ 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $Δ+ 2$ colours are enough for $Δ\geq 3$.
We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core.
Brooks' type theorems for coloring parameters of locally finite graphs and Konig's Lemma
本文研究了无限局部有限连通图着色参数的Brooks型定理，证明其在ZF中不可证但可由Konig引理推导，并提出了与Konig引理等价的新着色参数存在条件，同时给出了ZFC下无限图列表可辨别色数的上界。
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we study the total dominator chromatic number of some specific graphs.
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between total dominator chromatic number and chromatic number and total domination number.
Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
An assignment of numbers to the vertices of graph G is closed distinguishing if for any two adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the vertex v differs from the sum of labels of the vertices in the closed neighborhood of the vertex u unless they have the same closed neighborhood (i.e. N[u]=N[v]). The closed distinguishing number of G, denoted by dis[G], is the smallest integer k such that there is a closed distinguishing labeling for G using integers from the set[k].Also, for each vertex $v \in V(G)$, let L(v) denote a list of natural numbers …
A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, …
Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $ϕ: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge, $ϕ(e)\neq ϕ(e')$. The strong chromatic index of $G$, denoted as $χ'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the triangular prism then $χ_s'(G)\le 8$. In addition, they asked if the upper bound $8$ can be …
For a graph $G = (V(G), E(G))$, a dominating set $D$ is a vertex subset of $V(G)$ in which every vertex of $V(G) \setminus D$ is adjacent to a vertex in $D$. The domination number of $G$ is the minimum cardinality of a dominating set of $G$ and is denoted by $γ(G)$. A coloring of $G$ is a partition $C = (V_{1}, ... ,V_{k})$ such that each of $V_{i}$ in an independent set. The chromatic number is the smallest $k$ among all colorings $C = (V_{1}, ... ,V_{k})$ of $G$ and is denoted by $χ(G)$. A coloring $C = (V_{1}, ... ,V_{k})$ is said to be dominator if, for all $V_{i}$, every vertex $v \in V_{…
A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs. We also get bounds for the b-chromatic number of G -{x}, when G is a graph with large girth.
The Mycielskian construction, denoted $μ(G)$, takes a finite simple graph $G$ to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted $μ_t(G)$, takes $G$ to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of $μ(G)$ and $μ_t(G)$ in terms of the same parameters of $G$. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing.
The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest positive integer $k$ such that there exists a proper coloring for G with $k$ colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The $f$-chromatic vertex number of a $d$-regular graph $G$, denoted by $f(G)$, is the maximum number of dominant vertices of distinct colors in a proper coloring with $d+1$ colors. El Sahili and Kouider conjectured that $b(G)=d+1$ for any $d$-regular graph $G$ of girth 5. We study …
Let $G$ be a graph. We use $χ(G)$ and $ω(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices, and an $HVN$ is a $K_4$ together with one more vertex which is adjacent to exactly two vertices of $K_4$. Combining with some known result, in this paper we show that if $G$ is $(P_5, \textit{HVN})$-free, then $χ(G)\leq \max\{\min\{16, ω(G)+3\}, ω(G)+1\}$. This upper bound is almost sharp.
The Edge-Distinguishing Game
该研究提出图着色博弈 EDGe，以边区分色数界定玩家行动，并判定特定图与图族中的获胜方，同时结合博弈论和 Game of Cycles 的既有计算成果。
On Weakly Distinguishing Graph Polynomials
本文证明团多项式、独立多项式及满足有界退化度或树宽性质C的诱导子图生成函数和和谐色多项式均为弱区分图多项式。
A new algorithm to obtain the chromatic number of a finite, connected graph is proposed in this paper. The algorithm is based on contraction of non adjacent vertices.
The distinguishing chromatic number, $χ_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $φ$, of $G$, such that the only automorphism of $G$ that preserves all colors of $φ$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $χ_D(G)\le Δ+1$. We prove this conjecture.
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we examine the effects on $χ_d^t(G)$ when $G$ is modified by operations on vertex and edge of $G$.
We study Dohmen--Pönitz--Tittmann's bivariate chromatic polynomial $c_Γ(k,l)$ which counts all $(k+l)$-colorings of a graph $Γ$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_Γ(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_Γ(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynom…
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.
A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of color classes in a total dominator coloring of it. In [Total dominator chromatic number of a graph, submitted] the author initialed to study this number in graphs and obtained some important results. Here, we continue it in Mycieleskian graphs. We show that the total dominator chromatic number of the Mycieleskian of a graph $G$ belongs to between $χ_d^t(G)+1$ and $…
A tamed family of triangle-free graphs with unbounded chromatic number
构造了一个具有无界色数的无三角形图遗传类，其中每个非平凡图含不相邻孪生顶点或大小至多2的无边顶点割集，反面回答了Chudnovsky等人的问题，并证明twincut图是边临界的。
Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs
该研究针对偶数顶点且规模足够大的d-正则图，证明了顶点区分色指数和和区分色指数的相关结论，强化了已有结果并推动了领域发展。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。