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Register a new userCharacterizing the fullerene graphs with the minimum forcing number 3
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The minimum forcing number of a graph $G$ is the smallest number of edges simultaneously contained in a unique perfect matching of $G$. Zhang, Ye and Shiu cite{HDW} showed that the minimum forcing number of any fullerene graph was bounded below by $3$. However, we find that there exists exactly one excepted fullerene $F_{24}$ with the minimum forcing number $2$. In this paper, we characterize all fullerenes with the minimum forcing number $3$ by a construction approach. This also solves an open problem proposed by Zhang et al. We also find that except for $F_{24}$, all fullerenes with anti-forcing number $4$ have the minimum forcing number $3$. In particular, the nanotube fullerenes of type $(4, 2)$ are such fullerenes.
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In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)leq F(G)leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for bipartite graphs $G$, $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. In this paper, we solve the problem for general graphs and obtain that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or $K^+_{n,n}$ ($K_{n,n}$ with arbitrary additional edges in the same partite set). For a larger class of graphs $G$ with $F(G)=n-1$ we show that $G$ is $n$-connected and a brick (3-connected and bicritical graph) except for $K^+_{n,n}$. In particular, we prove that the forcing spectrum of each such graph $G$ is continued by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $lfloorfrac{n}{2}rfloor$ to $n-1$.
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ among all perfect matchings, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$ (see Lov{a}sz [20]). We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0leq kleq n-1$, and obtain that the number of edges is at most $n^2+2nk-k^2-k$ and characterize the extremal graphs as well. Conversely, we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of $F(G)$. Finally some open problems and conjectures are proposed.
Let $G$ be an $n$-vertex graph and let $L:V(G)rightarrow P({1,2,3})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where $n_i$ is the number of vertices in $G$ with list of size $i$ for $iin {2,3}$. As a corollary, we conclude that the 3-colorability of any graph $G$ with minimum degree at least 6 can be determined in $O(1.3196^{n-.5Delta(G)})$ time.
An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $kgeq3$. If $|E(G)|geq binom{n-m-(k+1-m)(delta+1)}{2} +(k+1-m)binom{delta+1}{2}+k+2$, then $pc(G)leq k$, where $m$ takes the value $t$ if $delta=1$ and $lfloor frac{k}{delta-1} rfloor$ if $deltageq2$. Furthermore, if $k=2$ and $delta=2$, %(i.e., $|E(G)|geq binom{n-5}{2} +7$) $pc(G)leq 2$, except $Gin {G_{1}, G_{n}}$ ($ngeq8$), where $G_{1}=K_{1}vee 3K_{2}$ and $G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_{1}vee (2K_{2}$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}vee (2K_{2}$) being joined. If $k=2$, $delta geq 3$, we conjecture $pc(G)leq 2$, where $m$ takes the value $1$ if $delta=3$ and $0$ if $deltageq4$ in the assumption.
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