All trees are six-cordial

For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing a labeling on the edges with edge-weights found by summing the labels on vertices incident to a given edge modulo $k$ so that each label appears on at most one more vertex than any other and each edge-weight appears on at most one more edge than any other. We prove that all trees are six-cordial by an adjustment of the test proposed by Hovey (1991) to show all trees are $k$-cordial.Comment: 16 pages, 12 figure

4-Prime cordiality of some cycle related graphs

Recently three prime cordial labeling behavior of path, cycle, complete graph, wheel, comb, subdivison of a star, bistar, double comb, corona of tree with a vertex, crown, olive tree and other standard graphs were studied. Also four prime cordial labeling behavior of complete graph, book, flower were studied. In this paper, we investigate the four prime cordial labeling behavior of corona of wheel, gear, double cone, helm, closed helm, butterfly graph, and friendship graph

Some results on integer cordial graph

An integer cordial labeling of a graph G(V, E) is an injective map f from V to or  as p is even or odd, which induces an edge labeling f*: E → {0, 1} defined by f*(uv) = 1 if f(u) + f(v) ≥ 0 is positive and 0 otherwise such that the number of edges labeled with1and the number of edges labeled with 0 differ atmost by 1. If a graph has integer cordial labeling, then it is called integer cordial graph. In this paper, we introduce the concept of integer cordial labeling and prove that some standard graphs are integer cordial

Path-cordial abelian groups

A labeling of the vertices of a graph by elements of any abelian group A induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph G to be A-cordial if it has such a labeling where the vertex labels and the edge labels are both evenly-distributed over A in a technical sense. His conjecture that all trees T are A-cordial for all cyclic groups A remains wide open, despite significant attention. Curiously, there has been very little study of whether Hovey’s conjecture might extend beyond the class of cyclic groups. We initiate this study by analyzing the larger class of finite abelian groups A such that all path graphs are A-cordial. We conjecture a complete characterization of such groups, and establish this conjecture for various infinite families of groups as well as for all groups of small order

Discussion On Some Important Topics in Graph Theory

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Mod/Resc Parsimony Inference

We address in this paper a new computational biology problem that aims at understanding a mechanism that could potentially be used to genetically manipulate natural insect populations infected by inherited, intra-cellular parasitic bacteria. In this problem, that we denote by \textsc{Mod/Resc Parsimony Inference}, we are given a boolean matrix and the goal is to find two other boolean matrices with a minimum number of columns such that an appropriately defined operation on these matrices gives back the input. We show that this is formally equivalent to the \textsc{Bipartite Biclique Edge Cover} problem and derive some complexity results for our problem using this equivalence. We provide a new, fixed-parameter tractability approach for solving both that slightly improves upon a previously published algorithm for the \textsc{Bipartite Biclique Edge Cover}. Finally, we present experimental results where we applied some of our techniques to a real-life data set.Comment: 11 pages, 3 figure