The genus polynomial of a graph is the generating polynomial for the number of nonequivalent embeddings of the graph on each orientable surface. In this paper, we address three questions on genus polynomials for wheel graphs: the computation of genus polynomials, the unimodality and the asymptotic normality of their coefficients. We derive an explicit formula for the genus polynomial of wheel graphs by combining methods of the joint tree model and characters theory, and then prove its real-rootedness. This stronger result implies the log-concavity, unimodality, and asymptotic normality of its coefficients. Thus, we confirm the unimodality conjecture for the genus distribution of wheel graphs and provide a positive answer to the asymptotic normality question posed by Zhang, Peng, and Chen (Adv. in Appl. Math. 127 (2021), 102175).

Let G be a graph of order n and P(G,x) be the chromatic polynomial of G. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that d^k/dx^k( ln[(-1)^n P(G, x)] ) < 0 holds for all k ≥ 2 and x ∈ (-∞, 0). We prove this conjecture for all k ≥ 2 and x≤ -10Δk, in which Δ is the maximum degree of G.

The mean color number of an -vertex graph , denoted by , is the average number of colors used in all proper -colorings of . For any graph and any vertex in , Dong (2003) conjectured that (1) ; (2) if is not an isolated vertex, then , where is a graph obtained from by deleting all but one of the edges incident to . We disprove these two conjectures by providing an infinite family of counterexamples.

Let G be a graph of order n with maximum degree Δ, and let P(G,x) denote its chromatic polynomial. We investigate several properties of P(G,x) related to its derivatives and higher-order derivatives. First, we study the monotonicity of P(G,x)/x^n. Dong proved that (x-1)^nP(G,x)≥ x^nP(G,x-1) for all real x≥ n. In particular, taking x=n establishes the Bartels-Welsh “shameful conjecture" that P(G,n)/P(G,n-1)>e. Fadnavis later showed that the same inequality holds for all real x≥ 36Δ^3/2. We improve this bound by proving that it already holds for all real x≥ 10Δ^3/2. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that d^k/dx^k( ln[(-1)^n P(G, x)] ) < 0 for all k ≥ 2 and x ∈ (-∞, 0). We establish this conjecture for all k ≥ 2 and x≤ -2.99Δk.

The authors explain and correct a mistake in [The thickness of amalgamations and Cartesian product of graphs, Discuss. Math. Graph Theory 37 (2017) 561-572]. The same mistake in [The thickness of the Cartesian product of two graphs, Canad. Math. Bull. 59 (2016) 705-720] is also corrected.

Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of t-bar visibility representation of undirected graphs. A t-bar visibility representation of a digraph G assigns each vertex at most t horizontal bars in the plane so that there is an arc xy in the digraph if and only if some bar for x "sees" some bar for y above it along an unblocked vertical strip with positive width. The visibility number b(G) is the least t such that G has a t-bar visibility representation. In this paper, we solve several problems about b(G) posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The DP-coloring is a generalization of the list coloring, introduced by Dvořák and Postle. Let ℋ=(L,H) be a cover of a graph G and P_DP(G,ℋ) be the number of ℋ -colorings of G. The DP color function P_DP(G,m) of G, introduced by Kaul and Mudrock, is the minimum value of P_DP(G,ℋ) where the minimum is taken over all possible m-fold covers ℋ of G. For the family of n-vertex connected graphs, one can deduce that trees maximize the DP color function, from two results of Kaul and Mudrock. In this paper we obtain tight upper bounds for the DP color function of n-vertex 2-connected graphs. Another concern in this paper is the canonical labeling in a cover. It is well known that if an m-fold cover ℋ of a graph G has a canonical labeling, then P_DP(G,ℋ)=P(G,m) in which P(G, m) is the chromatic polynomial of G. However the converse statement of this conclusion is not always true. We give examples that for some m and G, there exists an m-fold cover ℋ of G such that P_DP(G,ℋ)=P(G,m) , but ℋ has no canonical labelings. We also prove that when G is a unicyclic graph or a theta graph, for each m≥ 3 , if P_DP(G,ℋ)=P (G,m) , then ℋ has a canonical labeling.

A graph G is strongly even cycle decomposable if for every subdivision G′ of G with an even number of edges, the edges of G′ can be partitioned into cycles of even length. Máčajová and Mazák asked whether the line graph of a simple 2-connected cubic graph is strongly even cycle decomposable. A result of Seymour implies that the line graph of every 2-connected planar cubic graph is strongly even cycle decomposable. In this paper, we prove that the line graphs of simple 3-connected cubic graphs embedded in the projective plane or in the torus are strongly even cycle decomposable.

In 2015, Brown and Erey conjectured that every $2$-connected graph $G$ on $n$ vertices with chromatic number $k\geq 4$ has at most $(x-1)_{k-1}\big((x-1)^{n-k+1}+(-1)^{n-k}\big)$ proper $x$-colorings for all $x\geq k$. Engbers, Erey, Fox, and He proved this conjecture for $x=k$. In this paper, we prove Brown and Erey's conjecture under the condition that either the clique number of $G$ is $k$, or the independent number of $G$ is $2$.

The chromatic polynomial of a graph G is the polynomial function P ( G, m ) which counts the number of proper m -colorings of G . One classical problem in chromatic polynomials theory is to estimate bounds for the chromatic polynomials of graphs. In this note we consider the same problems in the context of DP-coloring which introduced by Dvoˇr´ak and Postle, obtain tight upper bounds for the DP color function of n -vertex 2-connected graphs.

For any graph G, the chromatic polynomial of G is the function P(G,m) which counts the number of proper m-colorings of G for each positive integer m. The DP color function P_DP(G,m) of G, introduced by Kaul and Mudrock in 2019, is a generalization of P(G,m) with P_DP(G,m)≤ P(G,m) for each positive integer m. Let P_DP(G)≈ P(G) (resp. P_DP(G)< P(G)) denote the property that P_DP(G,m)=P(G,m) (resp. P_DP(G,m)<P(G,m)) holds for sufficiently large integers m.It is an interesting problem of finding graphs G for which P_DP(G)≈ P(G) (resp. P_DP(G,m)<P(G,m)) holds. Kaul and Mudrock showed that if G has an even girth, then P_DP(G)<P(G) and Mudrock and Thomason recently proved that P_DP(G)≈ P(G) holds for each graph G which has a dominating vertex. We shall generalize their results in this article. For each edge e in G, let ℓ(e)=∞ if e is a bridge of G, and let ℓ(e) be the length of a shortest cycle in G containing e otherwise. We first show that if ℓ(e) is even for some edge e in G, then P_DP(G)<P(G) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that P_DP(G)≈ P(G) holds for every graph G that contains a spanning tree T such that for each e∈ E(G)∖ E(T), ℓ(e) is odd and e contained in a cycle C of length ℓ (e) with the property that ℓ(e')<ℓ(e) for each e'∈ E(C)∖ (E(T)∪{e}). Some open problems are proposed in this article.

A signed edge domination function (or SEDF) of a simple graph G=(V,E) is a function f:E→{1,−1} such that ∑e′∈N[e]f(e′)≥1 holds for each edge e∈E, where N[e] is the set of edges in G that share at least one endpoint with e. Let γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved for the two cases vodd(G)=0 and veven(G)=0, where vodd(G) (resp. veven(G)) is the number of odd (resp. even) vertices in G. This article proves Xu's conjecture for veven(G)∈{1,2}. We also show that for any simple graph G of order n, γs′(G)≤n+vodd(G)∕2 and γs′(G)≤n−2+veven(G) when veven(G)>0, and thus γs′(G)≤(4n−2)∕3. Our result improves the best current upper bound of γs′(G)≤⌈3n∕2⌉.

A graph is even cycle decomposable if its edges can be partitioned into cycles of even length. A graph G is strongly even cycle decomposable if every subdivision of G with an even number of edges is even cycle decomposable. Markström conjectured that for any simple 2-connected cubic graph G, its line graph L(G) is even cycle decomposable. Máčajová and Mazák further asked whether L(G) is strongly even cycle decomposable. In this paper, we resolve this question (as well as Markström's conjecture) in the affirmative for a class of cubic graphs. We prove that for a (not necessarily simple) 2-connected cubic graph G, if there exists a cycle C in G such that G−V(C) is a linear forest (i.e., a forest whose components are paths), then L(G) is strongly even cycle decomposable. Our main motivation for considering this class of graphs comes from a conjecture of Ash and Jackson that every cyclically 4-edge-connected cubic graph has a dominating cycle (i.e., a cycle whose deletion results in an independent set of vertices). If this conjecture is true, then our result will imply that the line graph of every cyclically 4-edge-connected cubic graph is strongly even cycle decomposable.

提出由边缘约束梁柱和双层斜向条型钢构件组成的钢网格墙结构,通过对1:2缩尺的T型钢和槽钢钢网格墙结构模型进行低周往复试验和有限元模拟,研究钢网格墙结构的抗震性能及受力机理,并基于工程实际对其内嵌钢网格布置进行了优化建议.结果表明:钢网格墙具有良好的刚度、延性和承载能力,且T型钢钢网格墙性能优于槽钢钢网格墙;受侧向力时中部型钢率先屈服,而后向两侧延展;试件破坏方式为单根条型钢构件与鱼尾板连接处断裂,其余型钢仍可承载;试件破坏时的层间位移角均大于1/50,试件在每级荷载循环后的强度退化系数均在0.92以上,结构受力性能稳定;有限元模型能较好地模拟钢网格墙性能,试件有限元模拟的峰值荷载最大误差为13％;内嵌型钢网格抗侧力及耗能的占比均高于48％,能够较好地与钢框架协调抗侧及耗能,对结构主体在侧向力作用下有一定保护作用;前期耗能主要由内嵌钢网格提供,随着型钢残余变形及受压屈曲,钢网格耗能占比逐渐减小;提出了两侧型钢鱼尾板连接处互不干扰原则、最小水平角距及最小竖直角距概念、均布约束准则,可通过调整最小水平角距实现合理的钢网格墙布置.

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. For the complete bipartite graph $K_{m,n}$, the lower bound $b(K_{m,n})\ge\lceil{\frac{mn+4}{2m+2n}}\rceil$ from Euler's Formula is well known. We prove that equality holds.

A t-bar visibility representation of a graph G assigns each vertex up to t horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least t such that G has a t-bar visibility representation is the bar visibility number of G, denoted by b(G). We show that if H is a spanning subgraph of G, then b(H)≤b(G)+1. It follows that b(G)≤⌈n∕6⌉+1 when G is an n-vertex graph. This improves the upper bound obtained by Chang et al. (2004).

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. In this paper, we present sharp lower and upper bounds for the thickness of the Kronecker product $G\times H$ of two graphs $G$ and $H$. We also give the exact thickness numbers for the Kronecker product graphs $K_n\times K_2$, $K_{m,n}\times K_2$ and $K_{n,n,n}\times K_2$.

The beam string structure (BSS) has been widely applied to public buildings (e.g. sports venues and exhibition centers) for its strong adaptability to architectural form and reasonable load bearing mechanism. However, most mathematical calculation methods for BSS are too complicated to be generally mastered by structural engineers, which limits the promotion and actual application. In this paper, two analytical calculation methods for the BSS are proposed based on displacement control objectives and work-energy principle. The computational formulas are then derived to calculate the member internal force and structural deformation. On this basis, the tension and static load tests and the finite element analytical method have been carried out to assess the calculation methods. The results of the tests and simulation are in good agreement with the analytical solution obtained by the computational formulas. Moreover, the formulas can be more appropriate with a greater beam span, proper rise–span and sag–span ratios (between 1/15 and 1/12) as well as more brace struts.

The $4$-girth-thickness $\theta(4,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least four whose union is $G$. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is $\big\lceil\frac{n+1}{2}\big\rceil$ except for $\theta(4,K_{1,1,1})=2$. And we also show that the $4$-girth-thickness of the complete graph $K_{10}$ is three which disprove the conjecture $\theta(4,K_{10})=4$ posed by Rubio-Montiel (Ars Math Contemp 14(2) (2018) 319).

图的厚度是指将该图分解为平面生成子图的最小数,它是衡量一个图可平面性的关键指标之一,研究一个图的厚度至关重要,它在超大规模集成电路和网络设计中有着重要应用.目前已经得到一部分图类的厚度的精确值,但完全二部图与完全三部图的厚度关系未完全得到,通过构造完全三部图K1,3p+1,6p+2的一个平面分解得到了完全三部图K1,n,2n的厚度,进而推出完全二部图Kn+1,2n与完全三部图K1,n,2n的厚度相等.