In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős' theorem and Moon-Moser's theorem, respectively. Let G k n be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in G k n (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in G k n . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.
Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is
defined as the number of colors of the edges incident to $v$. The color number
of $G$ is defined as the number of colors of the edges in $G$. A rainbow
triangle is one in which every pair of edges have distinct colors. In this
paper we give some sufficient conditions for the existence of rainbow triangles
in edge-colored graphs in terms of color degree, color number and edge number.
As a corollary, a conjecture proposed by Li and Wang (Color degree and
heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012)
1958--1964) is confirmed.Comment: Title slightly changed. 13 pages, to appear in European J. Combi
Let G be a graph on n vertices. An induced subgraph H of G is called heavy if there exist two nonadjacent vertices in H with degree sum at least n in G. We say thatIn this paper we characterize all connected graphs R and S other than P 3 (the path on three vertices) such that every 2-connected {R, S}-heavy graph is Hamiltonian. This extends several previous results on forbidden subgraph conditions for Hamiltonian graphs.
The 1-2-3 Conjecture asserts that, for every connected graph different from K 2 , its edges can be labeled with 1, 2, 3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graphs so that some pairs of their elements are distinguishable relatively to some parameter computed from the labeling. In this work, we investigate the consequences of labeling graphs as in the 1-2-3 Conjecture when it is further required to make the maximum resulting color as small as possible. In some sense, we aim at producing a number of colors that is as close as possible to the chromatic number of the graph. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permits to reduce the maximum color.
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