An equitable vertex \(k\)-coloring of a graph \(G\) is a proper vertex coloring with \(k\) colors such that the size of any two color classes differ by at most one. It was conjectured by Meyer in 1973 that every graph other than the complete graph \(K_n\) or an odd cycle \(C_{2m+1}\) admits an equitable vertex coloring with at most \(\Delta(G)\) colors, where \(\Delta(G)\) is the maximum degree of a vertex in graph \(G\). We show that the conjecture is true for middle graphs of some cycle-based graphs.

A questionnaire is a sequence of multiple choice questions aiming to collect data on a population. We define an abstract questionnaire as an ordered pair \((N,\mathcal{M})\), where \(N\) is a positive integer and \(\mathcal{M}=(m_0,m_1,\ldots,m_{N-1})\) is an \(N\)-tuple of positive integers, with \(m_i\), for \(i \in \mathbb{Z}_N\), as the number of possible answers to question \(i\). An abstract questionnaire may be endowed with a skip-list (which tells us which questions to skip based on the sequence of answers to the earlier questions) and a flag-set (which tells us which sequences of answers are of special interest). An FS-decision tree is a decision tree of an abstract questionnaire that also incorporates the information contained in the skip-list and flag-set. The main objective of this paper is to represent the abstract questionnaire using a directed graph, which we call an FS-decision digraph, that contains the full information of an FS-decision tree, but is in general much more concise. We present an algorithm for constructing a fully reduced FS-decision digraph, and develop the theory that supports it. In addition, we show how to generate all possible orderings of the questions in an abstract questionnaire that respect a given precedence relation.

In \(1973\), Harary and Palmer posed the problem of enumeration of labeled graphs on \(n \geq 1\) unisolated vertices and \(l \geq 0\) edges. In \(1997\), Bender et al. obtained a recurrence relation representing the sequence \(A054548\)(OEIS) of labeled graphs on \(n \geq 0\) unisolated vertices containing \(q \geq \frac{n}{2}\) edges. In \(2020\), Bhavale and Waphare obtained a recurrence relation representing the sequence of fundamental basic blocks on \(n \geq 0\) comparable reducible elements, having nullity \(l \geq \lfloor \frac{n+1}{2} \rfloor\). In this paper, we prove the equivalence of these two sequences. We also provide an edge labeling for a given vertex labeled finite simple graph.

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\) such that every edge \(e\in E\) belongs to at least one copy of a given subgraph \(H\) of \(G\). A bijection \(f:V\cup E\to \{1,2,\dots,|V|+|E|\}\) is called an \(H\)-supermagic labeling if the sum of labels of all vertices and edges of every copy of \(H\) is equal to the same number \(\mu\) and the vertices are labeled with the first \(|V|\) integers. A \(p\)-calendula graph \(Cal_{m,p[n]}\) consists of a cycle \(C_m\) with \(p\) copies of \(C_n\) amalgamated to each edge of \(C_m\). We generalize a previous result by Pradipta and Salman on 1-calendula graphs by providing \(C_n\)-supermagic labelings of \(Cal_{m,p[n]}\) for all \(m,n\geq3, p\geq 1\), and \(m\neq n\). The case of \(m=n, \ p>1\) remains open.

In this note, we present sufficient conditions involving the independence number, the minimum degree, and the maximum degree for some Hamiltonian properties of a graph.