Adjacency eigenvalues of graphs without short odd cycles

AbstractIt is known that solar wind velocity fluctuates regularly with a period of about 1.3 years. This periodicity (and other signals with periods near to 1.1 and 0.9 years) has also been observed in biological data. The variation is a temporary feature, mostly being observed in the early 1990s. Here, the occurrence of these periodic signals in solar and geomagnetic activity between 1932 and 2005 has been investigated. The signal with 1.3 year period is present in geomagnetic activity only in a short interval after 1990 and to a lesser extent around 1942. At other times the signal is very weak or not present at all. Other periods are much lower amplitude and appear only sporadically throughout the time investigated. A connection between these periods and solar cycles (e.g. different even or odd cycles) has not been proven. It is possible that there is a long-term periodicity in the occurrence of the 1.3 year period but the time series data available is insufficient to confirm this. There are no such periodicities in solar activity. In order to gain a greater understanding of these periodic signals, we should search for their origin in interplanetary space.

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Between 2- and 3-Colorability

The Electronic Journal of Combinatorics ◽

10.37236/4673 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 1

Author(s):

Alan Frieze ◽

Wesley Pegden

Keyword(s):

Positive Integer ◽

Random Graphs ◽

The Other ◽

Chromatic Numbers ◽

Other Hand ◽

Odd Cycles

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$. These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.

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A note on compact and compact circular edge-colorings of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.431 ◽

2008 ◽

Vol Vol. 10 no. 3 (Graph and Algorithms) ◽

Author(s):

Dariusz Dereniowski ◽

Adam Nadolski

Keyword(s):

Approximation Algorithm ◽

Decision Problem ◽

Edge Coloring ◽

Exact Algorithm ◽

Weighted Graphs ◽

Edge Colorings ◽

Odd Cycle ◽

International Audience ◽

Np Complete ◽

Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

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Ramsey Numbers of Trees Versus Odd Cycles

The Electronic Journal of Combinatorics ◽

10.37236/5731 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 1

Author(s):

Matthew Brennan

Keyword(s):

Positive Integer ◽

Constant Factor ◽

Linear Functions ◽

Ramsey Numbers ◽

Odd Cycle ◽

Odd Cycles

Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.

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Vertex-Transitive Direct Products of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6999 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Richard H. Hammack ◽

Wilfried Imrich

Keyword(s):

Direct Product ◽

Direct Products ◽

Odd Cycle ◽

Vertex Transitive ◽

Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

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Eigenvalues of graphs

10.31979/etd.q2np-mw87 ◽

2004 ◽

Author(s):

Wafa Yacoub

Keyword(s):

Eigenvalues Of Graphs

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On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

The Electronic Journal of Combinatorics ◽

10.37236/5173 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 8

Author(s):

Jakub Przybyło

Keyword(s):

Connected Graph ◽

Special Class ◽

Maximum Degree ◽

Minimum Degree ◽

Probabilistic Approach ◽

The Family ◽

Edge Colourings ◽

Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

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