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The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be textit{independence equivalent} if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has size 1, while the class can contain arbitrarily many graphs for even paths. We also prove that the independence equivalence class of every even cycle consists of two graphs when $nge 2$ except the independence equivalence class of $C_6$ which consists of three graphs. The odd case remains open, although, using irreducibility results from algebra, we were able show that for a prime $p geq 5$ and $nge 1$ the independence equivalence class of $C_{p^n}$ consists of only two graphs.
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We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)leq R_k(P_n)leq R_k(C_n)leq kn+o(n)$. The upper bound was recently improved by Sarkozy who showed that $R_k(C_n)leqleft(k-frac{k}{16k^3+1}right)n+o(n)$. Here we show $R_k(C_n) leq (k-frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 le j le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard sprinkling argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and passwords.
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper.
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