In this short note, using the method developed in [10] and [11], we construct a highly symmetrical, non-simple, attractive \(7\)-Venn diagram. This diagram has the minimum number of vertices, \(21\). The only similar two, published in [1] and [11], differ from ours in many ways. One of them was found by computer search [1]. Both of them are “necklace” type Venn diagrams (see [14] for definition), but ours is not.

A graph is a unit interval graph (respectively, an \(\tilde{n}\)-graph) if we can assign to each vertex an open interval of unit length (respectively, a set of \(n\) consecutive integers) so that edges correspond to pairs of intervals (respectively, of sets) that overlap. Sakai [14] and Troxell [18] provide a linear time algorithm to find the smallest integer \(n\) so that a unit interval graph is an \(\mathbb{A}\)-graph, for the particular case of reduced connected graphs with chromatic number \(3\). This work shows how to obtain such smallest \(n\) for arbitrary graphs, by establishing a relationship with the work by Bogart and Stellpflug [1] in the theory of semiorders.

For words of length \(n\), generated by independent geometric random variables, we consider the probability that these words avoid a given consecutive \(3\)-letter pattern. As a consequence, we count permutations in \(S_n\) avoiding consecutive \(3\)-letter patterns.

A mimeomatroid is a matroid union of a matroid with itself. We develop several properties of mimeomatroids, including a generalization of Rado’s theorem, and prove a weakened version of a matroid conjecture by Rota [2].

The well-known Marriage Lemma states that a bipartite regular graph has a perfect matching. We define a bipartite graph \(G\) with bipartition \((X,Y)\) to be semi-regular if both \(x \mapsto\) deg \(x,x \in X\) and \(y \mapsto\) deg \(y, y \in Y\) are constant. The purpose of this note is to show that if \(G\) is bipartite and semi-regular, and if \(|X| < |Y|\), then there is a matching which saturates \(|X|\). (Actually, we prove this for a condition weaker than semi-regular.) As an application, we show that various subgraphs of a hypercube have saturating matchings. We also exhibit classes of bipartite graphs, some of them semi-regular, whose vertices are the vertices of various weights in the hypercube \(Q_n\), but which are not subgraphs of \(Q_n\).

The sum graph of a set \(S\) of positive integers is the graph \(G^+(S)\) having \(S\) as its vertex set, with two vertices adjacent if and only if their sum is in \(S\). A graph \(G\) is called a sum graph if it is isomorphic to the sum graph \(G^+(S)\) of some finite subset \(S\) of \(N\). An integral sum graph is defined just as the sum graph, the difference being that \(S\) is a subset of \(Z\) instead of \(N\). The sum number of a graph \(G\) is defined as the smallest number of isolated vertices when added to \(G\) results in a sum graph. The integral sum number of \(G\) is defined analogously. In this paper, we study some classes of integral sum graphs.

We say that a graph \(F\) strongly arrows \((G,H)\) and write \(F \longmapsto (G,H)\) if for every edge-coloring of \(F\) with colors red and blue, a red \(G\) or a blue \(H\) occurs as an induced subgraph of \(F\). Induced Ramsey numbers are defined by \(r^*(G,H) = \min\{|V(F)| : F \longmapsto (G,H)\}\).
The value of \(r^*(G,H)\) is finite for all graphs, and good upper bounds on induced Ramsey numbers in general, and for particular families of graphs are known. Most of these results, however, use the probabilistic method, and therefore do not yield explicit constructions. This paper provides several constructions for upper bounds on \(r^*(G,H)\), including:\(r^*(C_n) = r^*(C_n,C_n) \leq c^{(logn)^2}\), \(r^*(T,K_n) \leq |T|n^{|T|log|T|}\), \(r^*(B,C_n) \leq |B|^{\lceil log n \rceil +4}\) ,where \(T\) is a tree, \(B\) is bipartite, \(K_n\) is the complete graph on \(n\) vertices, and \(C_n\) is a cycle on \(n\) vertices. We also have some new upper bounds for small graphs: \(r^*(K_3 + e) \leq 21\), and \(r^*(K_4 – e) \leq 46\).

An \(L(2,1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x)-f(y)|\geq 2\quad\text{if}\quad d_G(x,y)=1\) and \(|f(x)-f(y)|\geq 1\quad\text{if}\quad d_G(x,y)=2\). The \(L(2,1)\)-labeling problem is to find the smallest number \(\lambda(G)\) such that there exists an \(L(2,1)\)-labeling function with no label greater than \(\lambda(G)\). Motivated by the channel assignment problem introduced by Hale, the \(L(2,1)\)-labeling problem has been extensively studied in the past decade. In this paper, we study this concept for digraphs. In particular, results on ditrees are given.

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(\overline{f}: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(f(uv) = |f(u) – f(v)|\) .Let \(v_f(0),v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0),e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – vf(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).

A \(t\)-uniform homeomorph \(P_t(G)\) of \(G\) is the graph obtained by replacing all edges of \(G\) by vertex disjoint paths of length \(t\). In this paper we investigate the cordiality of \(P_t(G)\), when \(G\) itself is cordial. We find, wherever possible, a cordial labeling of \(P_t(G)\), whose restriction to \(G\) is the original cordial labeling of \(G\) and prove that for a cordial graph \(G\) and a positive integer \(t\), (1) \(P_t(G)\) is cordial whenever \(t\) is odd, (2) for \(t \equiv 2 \pmod{4}\) a cordial labeling \(g\) of \(G\) can be extended to a cordial labeling \(f\) of \(P_t(G)\) iff \(e_0\) is even, (3) for \(t \equiv 0 \pmod{4}\), a cordial labeling \(g\) of \(G\) can be extended to a cordial labeling \(f\) of \(P_t(G)\) iff \(e_1\) is even.

The domination graph \(dom(D)\) of a digraph \(D\) has the same vertex set as \(D\), and \(\{u,v\}\) is an edge if and only if for every \(w\), either \((u,w)\) or \((v,w)\) is an arc of \(D\). In earlier work we have shown that if \(G\) is a domination graph of a tournament, then \(G\) is either a forest of caterpillars or an odd cycle with additional pendant vertices or isolated vertices. We have also earlier characterized those connected graphs and forests of non-trivial caterpillars that are domination graphs of tournaments. We complete the characterization of domination graphs of tournaments by describing domination graphs with isolated vertices.