1,503 research outputs found

    Odd Harmonious Labeling of Some Graphs

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    The labeling of discrete structures is a potential area of research due to its wide range of applications. The present work is focused on one such labeling called odd harmonious labeling

    On the number of unlabeled vertices in edge-friendly labelings of graphs

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    Let GG be a graph with vertex set V(G)V(G) and edge set E(G)E(G), and ff be a 0-1 labeling of E(G)E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling ff \emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{partial vertex labeling} if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. Vertices that are incident to an equal number of edges of both labels we call \emph{unlabeled}. Call a procedure on a labeled graph a \emph{label switching algorithm} if it consists of pairwise switches of labels. Given an edge-friendly labeling of KnK_n, we show a label switching algorithm producing an edge-friendly relabeling of KnK_n such that all the vertices are labeled. We call such a labeling \textit{opinionated}.Comment: 7 pages, accepted to Discrete Mathematics, special issue dedicated to Combinatorics 201

    All trees are six-cordial

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    For any integer k>0k>0, a tree TT is kk-cordial if there exists a labeling of the vertices of TT by Zk\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 kk 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 kk-cordial.Comment: 16 pages, 12 figure

    On cordial labeling of hypertrees

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    Let f:V→Zkf:V\rightarrow\mathbb{Z}_k be a vertex labeling of a hypergraph H=(V,E)H=(V,E). This labeling induces an~edge labeling of HH defined by f(e)=∑v∈ef(v)f(e)=\sum_{v\in e}f(v), where the sum is taken modulo kk. We say that ff is kk-cordial if for all a,b∈Zka, b \in \mathbb{Z}_k the number of vertices with label aa differs by at most 11 from the number of vertices with label bb and the analogous condition holds also for labels of edges. If HH admits a kk-cordial labeling then HH is called kk-cordial. The existence of kk-cordial labelings has been investigated for graphs for decades. Hovey~(1991) conjectured that every tree TT is kk-cordial for every k≥2k\ge 2. Cichacz, G\"orlich and Tuza~(2013) were first to investigate the analogous problem for hypertrees, that is, connected hypergraphs without cycles. The main results of their work are that every kk-uniform hypertree is kk-cordial for every k≥2k\ge 2 and that every hypertree with nn or mm odd is 22-cordial. Moreover, they conjectured that in fact all hypertrees are 22-cordial. In this article, we confirm the conjecture of Cichacz et al. and make a step further by proving that for k∈{2,3}k\in\{2,3\} every hypertree is kk-cordial.Comment: 12 page
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