15-344/Classnotes for Tuesday September 15

DBN: Classes: 2015-16: 15-344 / Navigation Welcome to Math 344! Edits to the Math 344 web sites no longer count for the purpose of good deed points. # Week of... Notes and Links 1 Sep 14 About This Class, Day One Handout, Tuesday, Hour 3 Handout, Thursday, Tutorial 1 Handout 2 Sep 21 Tuesday, Tutorial 2 Page 1, Tutorial 2 Page 2, Thursday, HW1 3 Sep 28 Tuesday, Class Photo, Tutorial 3 Page 1, Tutorial 3 Page 2, Thursday, HW2 4 Oct 5 Tuesday, Drawing $n$-cubes, Thursday, HW3 5 Oct 12 Tuesday, Tutorial Handout, Thursday, HW4 6 Oct 19 Tuesday, Tutorial Handout, Thursday 7 Oct 26 Term Test on Tuesday, Dijkstra Handout,Thursday,HW5 8 Nov 2 Tuesday, Tutorial Handout, Thursday, HW6, Sunday November 8 is the last day to drop this class 9 Nov 9 Monday-Tuesday is UofT Fall Break, Thursday, HW7 10 Nov 16 Tuesday, Thursday, HW8 11 Nov 23 Tuesday, Thursday, HW9 12 Nov 30 Tuesday, Thursday, HW10 13 Dec 7 Tuesday, FibonacciFormula.pdf, semester ends on Wednesday - no class Thursday F Dec 11-22 The Final Exam Register of Good Deeds Add your name / see who's in!

We mostly went over Day One Handout today.

Dror's notes above / Students' notes below

Lecture Note for September 15

DEFINITION 1 Graph A graph $G=(V,E)$ is a set $V=\{a,b,...\}$ (usually finite, "vertices") along with a set $E=\{(ab),(bc),(bd),...\}$ ("edges") of unordered pairs of distinct elements of $V$.

DEFINITION 2 Incident If an edge $e=(ab)\in {E}$, we say that $e$ is incident to $a$ and $b$.

DEFINITION 3 N-valent In a graph $G=(V,E)$, a vertex $u\in {V}$ is called bivalent if it is incident to precisely two edges and n-valent if incident to precisely n edges, where $n=0,1,2,..$.

DEFINITION 4 Edge Cover An edge cover for graph $G=(V,E)$ is a subset $C\subset {V}$ such that every edge of $G$ incident to at least one vertex in $C$.

DEFINITION 5 Independent Let $G=(V,E)$ be a graph. A subset $I\subset {V}$ is called independent if whenever $a,b\in {I}$, then $(ab)\notin {E}$.

THEOREM 1 Edge covers are complementary to independent sets. In other words, $C\subset {V}$ is an edge cover if and only if the complementary subset $V-C$ is an independent set.

Proof $(\rightarrow )$ Assume $C$ is an edge cover. I assert that $I=V-C$ is independent. Indeed, if $e=(ab)\in {E}$, then since $C$ is an edge cover, either $a\in {C}\implies a\notin {I}$ or $b\in {C}\implies b\notin {I}$, which implies $e$ does not connect any two elements of $I$.

$(\leftarrow )$ Assume $I=V-C$ is independent. Pick any edge $e=(ab)\in {E}$. As $I$ is independent, $(ab)$ does not connect any two members of $I$. Hence, either $a\notin {I}\implies a\in {C}$ or $b\notin {I}\implies b\in {C}$, which implies $e$ is incident to an element of $C$. QED