The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Let $G$ be embeddable on a surface whose Euler characteristic $χ$ is as large as possible, and assume $χ\leq0$. Gagarin-Zverovich and Huang have recently found upper bounds of $b(G)$ in terms of the maximum degree $Δ(G)$ and the Euler characteristic $χ(G)=χ$. In this paper we prove a better upper bound $b(G)\leqΔ(G)+\lfloor t\rfloor$ where $t$ is the largest real root of the cubic equation $z^3 + z^2 + (3χ- 8)z + 9χ- 12=0$; this upper bound is asymptotically equivalent to $b(G)\leqΔ(G)+1+\lfloor \sqrt{4-3χ} \rfloor$. We also establish further improved upper bounds for $b(G)$ when the girth, order, or size of the graph $G$ is large compared with its Euler characteristic $χ$.
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. In this paper we present new upper bounds for b(G) in terms of girth, order and Euler characteristic.
A Roman dominating function on a graph $G$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The Roman domination number, $γ_R(G)$, of $G$ is the minimum of $Σ_{v\in V (G)} f(v)$ over such functions. The Roman bondage number $b_R(G)$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with Roman domination number not equal to $γ_R(G)$. In this paper we obtain upper bounds on $b_{R}(G)$ in terms of (a) the average degree and maximum degree, and (b) Euler characteristic, girth and maximum degr…
In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.
The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices la…
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $Δ(G)$ and embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$, $b(G)\leq\min\{Δ(G)+h+2,Δ+k+1\}$. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of $h$ and $k$. In this paper we establish an improved explicit upper bound for $b(G)$, using the Euler characteristic $…
The total domination number of a graph $G$ without isolated vertices is the minimum number of vertices that dominate all vertices in $G$. The total bondage number $b_t(G)$ of $G$ is the minimum number of edges whose removal enlarges the total domination number. This paper considers grid graphs. An $(n,m)$-grid graph $G_{n,m}$ is defined as the cartesian product of two paths $P_n$ and $P_m$. This paper determines the exact values of $b_t(G_{n,2})$ and $b_t(G_{n,3})$, and establishes some upper bounds of $b_t(G_{n,4})$.
The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $Δ(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{Δ(G)+h+2, Δ(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. An orientable surface $\mathbb{S}_h$ of genus $h$, $h \geq 0$, is obtained from the sphere $\mathbb{S}_0$ by adding $h$ handles. A non-orientable surface $\mathbb{N}_q$ of genus $q$, $q \geq 1$, is obtained from the sphere by adding $q$ crosscaps. The Euler characteristic of a surface is defined by $χ(\mathbb{S}_h) = 2 - 2h$ and $χ(\mathbb{S}_q)= 2-q$. Let $G$ be a connected graph of order $n$ which is 2-cell embedded on a surface $\mathbb{M}$ with $…
A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The bondage number of a nonempty graph $G$ is the size of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. In this note, we study the bondage number of binomial random graph $G(n,p)$. We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we…
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 stability, of a graph $G$ is denoted by $st_D(G)$ and is the minimum number of vertices whose removal changes the distinguishing number. We obtain a general upper bound $st_D(G) \leqslant \vert V(G)\vert -D(G)+1$, and a relationships between the distinguishing stabilities of graphs $G$ and $G-v$, i.e., $st_D(G)\leqslant st_D(G-v)+1$, where $v\in V(G)$. Also we study the edge distinguishing stability num…
For a graph $G$ let $γ(G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets,but $G-v$ has a unique minimum dominating set for each $v\in V(G)$. We show that each hypo-$\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $γ(G-v) < γ(G)$ for all $v \in V(G)$. We obtain a …
For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the sum $f (V) = \cup_{v\in V} f (v)$, and the minimum weight of a Roman dominating function on $G$ is the Roman domination number $γ_R(G)$ of $G$. The Roman bondage number $b_R(G)$ of $G$ is the minimum cardinality of all sets $F \subseteq E$ for which $γ_R(G - F) > γ_R(G)$. A graph $G$ is in the class $\mathcal{R}_{UVR}$ if the Roman domination number remains un…
We prove that among all flag homology $5$-manifolds with $n$ vertices, the join of $3$ circles of as equal length as possible is the unique maximizer of all the face numbers. The same upper bounds on the face numbers hold for $5$-dimensional flag Eulerian normal pseudomanifolds.
We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's Theorem. If the alternating sum and the number of…
We prove that among all flag 3-manifolds on $n$ vertices, the join of two circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and $\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and find all maximizers of the fa…
Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cove…
The maximum number of vertices in a graph of maximum degree $Δ\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(Δ-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of $(2+o(1))(Δ-1)^{\lfloor k/2\rfloor}$ for a fixed $k$. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus $g$ behave like trees, in the sense that, for large $Δ$, such graphs have orders bounded from above by \[begin{cases} c(g+1)(Δ-1)^{\lfloor k/2\rfloor} & \text{if $k$ is even} c(g…
Computing and Bounding the Number of Eulerian Orientations for Certain Classes of $4$-Regular Graphs
Department of Informatics & Telecommunications, National & Kapodistrian University of Athens（雅典国立卡波迪斯特里阿斯大学信息与通信系）；Athens University of Economics and Business（雅典经济与商业大学）
Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an $O(1)$-approximation is not ruled out, the currently best-known upper bound is a trivial $O(n/g)$-approximation that follows from bounds on the Euler characteristic. In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we …
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers. It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor (Grohe, JACM 2012). However, the bounds that can be derived from this general result ar…
In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $γ_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $π^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's cons…
We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2π\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for the number of edges in terms of $n$ and the number of non-triangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.
