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Register a new userThe study of odd graceful and odd strongly harmonious for bipartite graph
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In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is bipartite odd graceful graph, then G is odd strongly harmonious graph.
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The concept of graceful labels was proposed by Rosa, scholars began to study graceful labels of various graphs and obtained relevant results.Let the graph is a bipartite graceful graph, we have proved some graphs are graceful labeling in this paper.
We prove that every family of (not necessarily distinct) odd cycles $O_1, dots, O_{2lceil n/2 rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.
Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a well-structured class of Stanley sequences with two main parameters: the character $lambda(A)$ and the repeat factor $rho(A)$. Rolnick conjectured that for every $lambda in mathbb{N}_0backslash{1, 3, 5, 9, 11, 15}$, there exists an independent Stanley sequence $S(A)$ such that $lambda(A) =lambda$. This paper demonstrates that $lambda(A) otin {1, 3, 5, 9, 11, 15}$ for any independent Stanley sequence $S(A)$.
We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result.
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