In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs. Extremal graphs achieving the maximum values are also given.

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The number of maximum independent sets in graphs

/pdf/independence-polynomials-of-circulant-graphs-2ydibivtem.pdf

Independence polynomials of circulant graphs

https://www.mscs.dal.ca/~hoshino/music.pdf

Independence polynomials of circulants with an application to music

We consider the number of vertex independent sets $i(G)$. In general, the problem of determining the value of $i(G)$ is $NP$-complete. We present several upper and lower bounds for $i(G)$ in terms of order, size or independence number. We obtain improved bounds for $i(G)$ on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

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Bounds on the number of vertex independent sets in a graph

A colorful path in a graph $G$ is a path with $\chi(G)$ vertices whose colors are different. A $v$-colorful path is such a path, starting from $v$. Let $G\neq C_7$ be a connected graph with maximum degree $\Delta(G)$. We show that there exists a $(\Delta(G)+1)$-coloring of $G$ with a $v$-colorful path for every $v\in V(G)$. We also prove that this result is true if one replaces $(\Delta(G)+1)$ colors with $2\chi(G)$ colors. If $\chi(G)=\omega(G)$, then the result still holds for $\chi(G)$ colors. For every graph $G$, we show that there exists a $\chi(G)$-coloring of $G$ with a rainbow path of length $\lfloor\chi(G)/2\rfloor$ starting from each $v \in V(G)$.

/pdf/colorful-paths-in-vertex-coloring-of-graphs-1m14xksi6y.pdf

Colorful Paths in Vertex Coloring of Graphs

In this note we give simple proofs of two results which concerns paths representing all colors in optimal proper vertex-colorings of graphs. One result, due to Fung [2], is about the existence of colorful paths. The other one, due to Li [4], is about the existence of paths starting at any given vertex and representing all colors. Besides, we propose some open problems.

Simple Proofs of Results on Paths Representing All Colors in Proper Vertex-Colorings

In this paper we prove the following result. Let m1 ≥ m2 ≥ … ≥ m1 be nonnegative integers. A necessary and sufficient condition for the complete graph Kn to be decomposed into stars Sm1, Sm2, …, Sm1 is . © 1996 John Wiley & Sons, Inc.

A necessary and sufficient condition for the star decomposition of complete graphs

Isomorphic star decompositions of multicrowns and the power of cycles

Denote bymi(G) the number of maximal independent sets ofG. This paper studies the setS(k) of all graphsG withmi(G) = k and without isolated vertices (exceptG ? K 1) or duplicated vertices. We determineS(1), S(2), andS(3) and prove that |V(G)| ≤ 2 k?1 +k ? 2 for anyG inS(k) andk ? 2; consequently,S(k) is finite for anyk.

Chiang Lin

Papers

Simple Proofs of Results on Paths Representing All Colors in Proper Vertex-Colorings

A necessary and sufficient condition for the star decomposition of complete graphs

Isomorphic star decompositions of multicrowns and the power of cycles

A finiteness theorem for maximal independent sets

Balanced star decompositions of regular multigraphs and λ-fold complete bipartite graphs