Let $S$ be a finite set of positive integers. A mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K. Wang, The smallest one-realization of a given set \uppercase\expandafter{\romannumeral2}, Discrete Math. 312 (2012) 2946--2951], we determined the minimum number of vertices of a 3-uniform bi-hypergraph which is a one-realization of $S$. In this paper, we generalize this result to $r$-uniform bi-hypergraphs.
In [The smallest one-realization of a given set, Electronic J. Combin. 19 (2012), $\sharp$P19], we determined the minimum number of vertices of one-realizations of a given finite set $S$, and constructed the corresponding mixed hypergraphs. In this paper, by finding some of their spanning sub-hypergraphs, we determine the minimum number of ${\cal D}$-deges (resp. ${\cal C}$-edges) of one-realizations of $S$. As a result, we partially solve an open problem proposed by Tuza and Voloshin in 2008.
For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In this paper, we determine the minimum size of 3-uniform bi-hypergraphs which are one-realizations of a given set $S$. As a result, we partially solve an open problem proposed by Bujt$\acute{\rm a}$s and Tuza in 2008.
For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. \cite{Jiang} showed that the minimum number of vertices of realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Kr$\acute{\rm a}$l \cite{Kral} proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we improve Kr$\acute{\rm a}$l's result, and determine the size of …
In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $σ(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $σ(S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $σ(H, n)$. We show $σ(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $σ(C_{2m+2}, n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$.
The $3$-polytopes are planar, $3$-connected graphs. A classical question is, for $r\geq 3$, is the $2(r-1)$-gonal prism $K_2\times C_{2(r-1)}$ the unique $3$-polytope of graph radius $r$ and smallest size? Under some extra assumptions, we answer this question in the positive.
Identifying graphs with extremal properties is an extensively studied topic in spectral graph theory. In this paper, we study the log-concavity of a type of iteration sequence related to the $α$-normal weighted incidence matrices which is presented by Lu and Man for computing the spectral radius of hypergraphs. By using results obtained about the sequence and the method of some edge operations, we will characterize completely extremal k-graphs with the smallest spectral radius among bicyclic hypergraphs with given size.
For a sequence $d$ of non-negative integers, let ${\cal F}(d)$ be the set of all forests whose degree sequence is $d$. We present closed formulas for $γ_{\max}^{\cal F}(d)=\max\{ γ(F):F\in {\cal F}(d)\}$ and $α_{\min}^{\cal F}(d)=\min\{ α(F):F\in {\cal F}(d)\}$ where $γ(F)$ and $α(F)$ are the domination number and the independence number of a forest $F$, respectively.
The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques…
In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\sf smallest defining set} (of vertex coloring) and its cardinality, the {\sf defining number}, is denoted by $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al \cite{rkgraph} proved that, for a given $k$ and for al…
We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of partitions of i into distinct parts with even rank minus the number with odd rank, for which there is an almost closed formula by Andrews, Dyson and Hickerson. This method of calculating the number of partitions of n with a given parity of the smallest part is practical and efficient.
Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices a…
For a sequence $d$ of non-negative integers, let ${\cal G}(d)$ and ${\cal F}(d)$ be the sets of all graphs and forests with degree sequence $d$, respectively. Let $γ_{\min}(d)=\min\{ γ(G):G\in {\cal G}(d)\}$, $α_{\max}(d)=\max\{ α(G):G\in {\cal G}(d)\}$, $γ_{\min}^{\cal F}(d)=\min\{ γ(F):F\in {\cal F}(d)\}$, and $α_{\max}^{\cal F}(d)=\max\{ α(F):F\in {\cal F}(d)\}$ where $γ(G)$ is the domination number and $α(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $α_{\max}(d)$ can be determined in polynomial time. We establish the existence…
For a finite group $G$, denote by $α(G)$ the minimum number of vertices of any graph $Γ$ having $\text{Aut}(Γ)\cong G$. In this paper, we prove that $α(G)\leq |G|$, with specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups. Additionally, we compute $α(G)$ for the groups $G$ such that $α(G)> |G|$ where the value $α(G)$ was previously unknown.
Given a graph $H$, a graph $G$ is called $H$-critical if $G$ does not admit a homomorphism to $H$, but any proper subgraph of $G$ does. Observe that $K_{k-1}$-critical graphs are the standard $k$-(colour)-critical graphs. We consider questions of extremal nature previously studied for $k$-critical graphs and generalize them to $H$-critical graphs. After complete graphs, the next natural case to consider for $H$ is that of the odd-cycles. Thus, given integers $\ell$ and $k$, $\ell\geq k$, we ask: what is the smallest order of a $C_{2\ell +1}$-critical graph of odd-girth at least $2k+1$? Denotin…
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.
For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $χ_{\min}(d)$ and the largest chromatic number $χ_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then $χ_{\min}(d)\leq \max\left\{ 3,d_1-\frac{n+1}{4d_1}+4\right\}$, and, if $\sqrt{n+\frac{1}{4}}-\frac{1}{2}>d_1\geq d_n\geq 1$, then $χ_{\max}(d)=\max\limits_{i\in [n]}\min\left\{ i,d_i+1\right\}$. For a given degree sequence $d$ with bounded entries, we show that $χ_{\min}(d)$, $χ_{\max}(d)$, and also the smallest independence number $α_{\min}(d)$ among all graphs wit…
Given any connected poset $P$, we give a simple realization of Galashin's poset associahedron $\mathscr{A}(P)$ as a convex polytope in $\mathbb{R}^P.$ The realization is inspired by the description of $\mathscr{A}(P)$ as a compactification of the configuration space of order-preserving maps $P \to \mathbb{R}.$ In addition, we give an analogous realization for Galashin's affine poset cyclohedra.
Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.
Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by $n-$cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；ELTE: Eötvös Loránd University, Mathematics Institute, Department of Computer Science（厄特沃什·罗兰大学数学研究所计算机科学系）
We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear…
Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers. We prove that c(G)\le (1/4+o(1))n^2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch, and Tuza [Discrete Mathematics, 144 (1995) 3-10]. Also, we show that, for every e>0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73-e.
The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one w…
One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em swap} operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of fast …
This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitriangulations based on their stars, which play the same role as triangles of triangulations. This study leads to interpret multitriangulations by duality as pseudoline arrangements with contact points cove…
A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G. Ringel [J. Comb. Theory, Ser. B, 1989, 46, 84-95].
