Try to Imitate This La Tex File for This Assignment

2006Spring CS157B Assignment 2 Prof. Sin-Min Lee Section 2

Due: February 23, 2006 Thursday

Try to imitate this La TeX file for this assignment

Example 1. \documentclass{article} % 37th SE Combo Conf., Boca Raton, 3/6-3/10/06 \oddsidemargin 0in \textwidth 6in \textheight 5in \pagestyle{empty}

\begin{document}

\begin{center}\Large On Edge-Magic $(p,3p-1)$-Graphs \end{center} \begin{center} Dharam Chopra* \\ Wichita State University \\ \ \\ Harris Kwong \\ State University of New York at Fredonia \\ \ \\ Sin-Min Lee \\ San Jose State University \end{center}

\bigskip\noindent A $(p,q)$-graph $G$ in which the edges are labeled $1,2,3,\ldots,q$ so that the vertex sums are constant, is called \textbf{\emph{supermagic}}. If the vertex sum modulo $p$ is a constant, then $G$ is called \textbf{\emph{edge-magic}}. Lee, Seah and Tan introduced the concept of edge-magic graphs, which is a weaker concept than the supermagic. In this paper, we investigate edge-magic $(p,q)$-graphs where $q$ are of the form $3p-1$. Trees, cycles and unicyclic graphs are not edge-magic. However, using the construction of edge-splitting extension on trees, cycles and unicyclic graphs we provide infinite families of edge-magic graphs.

\bigskip\noindent \textbf{Keywords}: edge-magic, supermagic, edge-splitting extension, tree, unicyclic graphs, partition, amalgamation.

\end{document}

If you run the LaTeX, you will get the following result in pdf format. Example 2.

\documentclass{article} \usepackage{graphics,latexsym}

\oddsidemargin 1.00in \textwidth 4.500truein \topmargin 0.40in \textheight 7.125truein \tolerance 3000

\newtheorem{thm}{Theorem}[section] \newtheorem{lem}{Lemma} \renewcommand{\thelem}{} \newtheorem{cor}[thm]{Corollary} \def\qed{\hfill$\Box$}

\begin{document} \title{\textbf{On $Q(a)P(b)$-Super Edge-Graceful Wheels}} \author{% \textbf{Sin-Min Lee} \\ \textbf{Siao Ling Song} \\ Department of Computer Science\\ San Jose State University\\ San Jose, California 95192 \and \textbf{Linda Vald\'es} \\ Department of Mathematics\\ San Jose State University\\ San Jose, California 95192}

% \date{} \date{\textbf{In Memory of Professor Amy Rocha}} \maketitle \begin{abstract} Let $a,b$ be two positive integers. For the graph $G$ with vertex set $V(G)$ and edge set $E(G)$ with $p=|V(G)|$ and $q=|E(G)|$, we define two sets $Q(a)$ and $P(b)$ as follows: \[ \displaylines{ Q(a) = \cases{ \{\pm a,\pm(a+1),\ldots,\pm(a+(q-2)/2)\} & if $q$ is even, \cr \{0\}\cup\{\pm a,\pm(a+1),\ldots,\pm(a+(q-3)/2)\} & if $q$ is odd, \cr} \cr P(b) = \cases{ \{\pm b,\pm(b+1),\ldots,\pm(b+(p-2)/2)\} & if $p$ is even, \cr \{0\}\cup\{\pm b,\pm(b+1),\ldots,\pm(b+(p-3)/2)\} & if $p$ is odd. \cr} } \] The graph $G$ is said to be $Q(a)P(b)$-super edge-graceful (in short $Q(a)P(b)$-SEG) if there exists a \emph{function pair} $(f,f^{+})$ which assigns integer labels to the vertices and edges, that is, $f^{+}:V(G)\rightarrow P(b),$ and $f:E(G)\rightarrow Q(a)$, such that $f^{+}$is onto $P(b)$ and $f$ is onto $Q(a)$, and $f^{+}(u)=\sum_{(u,v)\in E(G)} f(u,v)$. We investigate $Q(a)P(b)$-super edge-gracefulness for wheel graphs. \end{abstract}

\bigskip\noindent\textbf{Keywords:} edge-graceful, super edge-graceful, $Q(a)P(b)$-SEG, wheel graphs. \par\bigskip\noindent \textbf{2000 MR Classification:} 05C78.

% ======\section{Introduction} % ======

If $G$ is a $(p,q)$ graph in which the edges are labeled $1,2,3,\ldots,q$ so that the vertex sums defined by $h^+(u) = \sum_{(u,v)\in E} h(u,v)$ (mod~$p$) are distinct, then $G$ is called \textbf{edge-graceful}. Edge-graceful labeling of graphs was introduced by S.P.~Lo \cite{r20} in 1985. A necessary condition of edge-gracefulness is \[ q(q+1)\equiv \frac{p(p-1)}{2} \pmod{p}. \] This condition may be more practically stated as $q(q+1) \equiv 0$ or $p/2$ (mod~$p$), depending on whether $p$ is odd or even.

% ======% \section*{References} % ======

\begin{thebibliography}{99}

\bibitem{r1} D. Chopra and Sin-Min Lee, On $Q(a)P(b)$-super edge-graceful graphs, \textsl{J. Combin.\ Math. Combin. Comput.}, to appear.

\bibitem{r2} J.A. Gallian, A dynamic survey of graph labeling, \textsl{Electronic J. Combin.} (2001), \#DS6, 1--79. \bibitem{r3} Jonathan Keene and Andrew Simoson, Balanced strands for asymmetric, edge-graceful spiders, \textsl{Ars Combin.} \textbf{42} (1996), 49--64.

\bibitem{r4} Q. Kuan, Sin-Min Lee, J. Mitchem, and A.K. Wang, On edge-graceful unicyclic graphs, \textsl{Congr. Numer.} \textbf{61} (1988), 65--74.

\bibitem{r5} H. Kwong and Sin-Min Lee, On $Q(a)P(b)$-super edge-gracefulness of hypercubes, \textsl{J. Combin.\ Math. Combin. Comput.}, to appear.

\bibitem{r6} Li Min Lee, Sin Min Lee, and G. Murty, On edge-graceful labelings of complete graphs --- solutions of Lo's conjecture, \textsl{Congr. Numer.} \textbf{62} (1988), 225--233.

\bibitem{r7} Sin-Min Lee, A conjecture on edge-graceful trees, \textsl{Scientia, Ser.~A} \textbf{3} (1989), 45--57.

\bibitem{r8} Sin-Min Lee, New directions in the theory of edge-graceful graphs, in \textsl{Proc.\ 6th Caribbean Conference Combinatoric and Computing} (1991) 216--231.

\bibitem{r9} Sin-Min Lee, E. Chen, E.R.Yera and Ling Wang, On super edge-graceful $(p,p+1)$-graphs, \textsl{Congr. Numer.} \textbf{171} (2004), 51--65.

\bibitem{r10} Sin-Min Lee, K.J. Chen and Yung-Chin Wang, On the edge-graceful spectra of the cycles with one chord and dumbbell graphs, \textsl{Congr. Numer.} \textbf{170} (2004), 171--183.

\bibitem{r11} Sin-Min Lee, Peining Ma, Linda Valdes, and Siu-Ming Tong, On the edge-graceful grids, \textsl{Congr. Numer.} \textbf{154} (2002), 61--71.

\bibitem{r12} Sin Min Lee and Eric Seah, Edge-graceful labelings of regular complete $k$-partite graphs, \textsl{Congr. Numer.} \textbf{75} (1990), 41--50.

\bibitem{r13} Sin Min Lee and Eric Seah, On edge-gracefulness of the composition of step graphs with null graphs, \textsl{Combinatorics, Algorithms, and Applications in Society for Industrial and Applied Mathematics} (1991), 326--330.

\bibitem{r14} Sin Min Lee and Eric Seah, On the edge-graceful $(n,kn)$-multigraphs conjecture, \textsl{J. Combin.\ Math. Combin. Comput.} \textbf{9} (1991), 141--147. \bibitem{r15} Sin-Min Lee, E. Seah and S.P. Lo , On edge-graceful 2-regular graphs, \textsl{J. Combin.\ Math. Combin. Comput.} \textbf{12} (1992), 109--117.

\bibitem{r16} Sin-Min Lee, E. Seah, Siu-Ming Tong, On the edge-magic and edge-graceful total graphs conjecture, \textsl{Congr. Numer.} \textbf{141} (1999), 37--48.

\bibitem{r17} Sin-Min Lee, E. Seah and P.C. Wang , On edge-gracefulness of the $k$th power graphs, \textsl{Bulletin of the Institute of Math, Academia Sinica} \textbf{18} (1990), 1--11.

% \bibitem{r18} % Sin Min Lee, Eric Seah and S.K. Tan, On edge-magic graphs, % \textsl{Congr. Numer.} \textbf{86} (1992), 179--191.

% \bibitem{r19} % Sin-Min Lee, E. Seah, Siu-Ming Tong, On the edge-magic and % edge-graceful total graphs conjecture, \textsl{Congr. Numer.} % \textbf{141} (1999), 37--48.

\bibitem{r20} S.P. Lo, On edge-graceful labelings of graphs, \textsl{Congr. Numer.} \textbf{50} (1985), 231--241.

\bibitem{r21} Peng Jin. and Li W., Edge-gracefulness of $C_m\times C_n$, in \textsl{Proceedings of the Sixth Conference of Operations Research Society of China}, (Hong Kong: Global-Link Publishing Company), Changsha, October 10--15, 2000, 942--948.

\bibitem{r22} J. Mitchem and A. Simoson, On edge-graceful and super-edge-graceful graphs, \textsl{Ars Combin.} \textbf{37} (1994), 97--111.

\bibitem{r23} A.Riskin and S. Wilson, Edge graceful labelings of disjoint unions of cycles, \textsl{Bull.\ Instit.\ Combin.\ Appl.} \textbf{22} (1998), 53--58.

\bibitem{r24} Karl Schaffer and Sin-Min Lee, Edge-graceful and edge-magic labelings of Cartesian products of graphs, \textsl{Congr. Numer.} \textbf{141} (1999), 119--134.

\bibitem{r25} W.C. Shiu, Sin-Min Lee and K. Schaffer, Some $k$-fold edge-graceful labelings of $(p,p-1)$-graphs, \textsl{J. Combin.\ Math. Combin. Comput.} \textbf{38} (2001), 81--95.

% \bibitem{r26} % Yihui Wen, Sin-Min Lee, and Hugo Sun, On the supermagic edge-splitting % extension of graphs, \textsl{Ars Combin.}, to appear. \bibitem{r27} Yihui Wen, Sin-Min Lee and Siu-Ming Tong, On the edge-graceful edge-splitting extension of stars, \textsl{J. Combin.\ Math. Combin. Comput.}, to appear.

\bibitem{r28} S. Wilson and A. Riskin, Edge-graceful labelings of odd cycles and their products, \textsl{Bull.\ Instit. Combin.\ Appl.} \textbf{24} (1998), 57--64.

\end{thebibliography} \end{document}

Section2 Exercise: On The Edge-Magic (p,2p-1)-Graphs

Sin-Min Lee and Brian Chan Department of Computer Science San José State University San José, California 95192 U.S.A.

Siu-Ming Tong Department of Computer Engineering Northwestern Polytechnic University Fremont, California 94539 U.S.A.

ABSTRACT A (p,q)-graph G in which the edges are labeled 1,2,3,...q so that the vertex sums are constant, is called supermagic. If the vertex sum mod p is a constant, then G is called edge-magic. Lee, Seah and Tan [11] introduced the concept of edge-magic graphs, which is a weaker concept than the supermagic. In this paper, we investigate edge-magic (p,q)-graphs where q = 2p-1. Using the construction of edge-splitting extension on trees, cycles and unicyclic graphs we provide infinitely many families of edge- magic graphs.

Keywords: edge-magic, supermagic, edge-splitting extension, tree, unicyclic graphs, partition, amalgamation.

M.R. Classification: 05C78

1.Introduction. A (p,q)-graph G, in which the edges are labeled 1,2,3,...q so that the vertex sums are constant, is called supermagic. B. M. Stewart [21,22] showed that the complete graphs K3, K4, K5 are not supermagic and for n>5, Kn is supermagic if and only if n 0 (mod 4). Hartsfield and G. Ringel [4] provided some classes of supermagic graphs. Ho and Lee [5] extended the result of Stewart to regular complete k-partite graphs. Recently, Shiu, Lam and Cheng [15] considered a class of supermagic graphs which are disjoint unions of K3,3. Shiu, Lam and Lee [17] proved that the composition of regular supermagic graph with a null graph is always supermagic. In [6] supermagic regular multipartite graphs and supermagic cubes are characterized.

Definition 1. If G is a (p,q)-(multi-)graph in which the edges are labeled 1,2,3,...q so that the vertex sums are constant, mod p, then G is called an edge-magic graph (EM graph).

Example 1. Figure 1 shows a non-supermagic graph G1 with 6 vertices and 8 edges that is EM with different constant sums (mod 6). References

[1] M.Baca, I. Hollander, Ko-Wei Lih, Two classes of super-magic graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 23 (1997),113-120.

[2] R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle , On super edge-magic graphs, Ars Combinatoria 64 (2002), 81-96.

[3] J.A. Gallian, A dynamic survey of graph labeling, The Electronic J. of Combin. (2001), # DS6, 1-144.

[4] N. Hartsfield and G. Ringel, Supermagic and Antimagic Graphs, Journal of Recreational Mathematics, 21 (1989), 107-115.

[5] Y.S. Ho and Sin-Min Lee, An initial result of supermagicness of complete k-partite graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 39 (2001) 3 - 17

[6] J. Ivanco, On supermagic regular graphs, Mathematica Bohemica 125 (2000),99-114.

[7] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canada Math. Bull. 13 (1970), 451-461.

[8] A. Kotzig and A. Rosa, Magic valuations of complete graphs, Publications du Centre de Recherches Mathematiques Universite de Montreal 175 (1972)

[9] Sin-Min Lee , H.K. Ng and Yihui Wen, On the edge-magic indices of (v, v+1)- graphs, to appear in Utilita Mathematica.

[10] Sin-Min Lee, W.M. Pigg and T.J. Cox, On edge-magic cubic graphs conjecture, Congressus Numerantium, 105 (1994), 214-222. [11] Sin Min Lee, Eric Seah and S.K. Tan, On edge-magic graphs, Congressus Numerantium, 86 (1992), 179-191.

[12] Sin-Min Lee, E. Seah, Siu-Ming Tong, On the edge-magic and edge-graceful total graphs conjecture, Congressus Numerantium 141(1999), 37-48.

[13] Sin-Min Lee, Wang Ling, and Yihui Wen, On edge-magic cubic multi graphs and graphs, Congressus Numerantium 165 (2003) 145-160.

[14]Karl Schaffer and Sin Min Lee, Edge-graceful and edge-magic labelings of Cartesian products of graphs, Congressus Numerantium 141 (1999), 119-134,

[15] W.C. Shiu, P.C.B. Lam and H.L. Cheng, Supermagic labeling of an s-duplicate of Kn,n, Congressus Numerantium, 146(2000) 119-124.

[16] W.C. Shiu, P.C.B. Lam and Sin-Min Lee, Edge-magicness of the composition of a cycle with a null graph, Congressus Numerantium, 132(1998), 9-18.

[17] W.C. Shiu, P.C.B. Lam and Sin-Min Lee, On a construction of supermagic graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 42(2002),147-160

[18] W.C. Shiu, F.C.B. Lam and Sin-Min Lee, Edge-magic index sets of (p,p)-graphs, Congressus Numerantium 137(1999), p.97-107

[19]W.C. Shiu, F.C.B. Lam and Sin-Min Lee, Edge-magic indices of (n,n-1)-graphs, Electron. Notes in Discrete Mathematics, 11, July (2002)

[20]W.C. Shiu and Sin-Min Lee, Some edge-magic cubic graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 40 (2002),115-127.

[21] B.M. Stewart, Magic graphs, Canadian Journal of Mathematics, 18(1966), 1031-1059.

[22] B.M. Stewart, Supermagic complete graphs, Canadian Journal of Mathematics, 19(1967), 427-438.

[23] Yihui Wen, Sin-Min Lee and Hugo Sun, On the supermagic edge-splitting extension of graphs, to appear in JCMCC.

Please submit the assignment to the following email address: Please email the text in pdf format of document and in the body of the email contain the copy of the LaTeX format of the document.

Section 2: f157 b [email protected]