LaTeX vs MathPix image — Introduction to Graph and Hypergraph Theory (Vitaly I. Voloshin) (Z-Library)

G=(X, E) .

\sum_{i=1}^{n} d\left(x_{i}\right)=2 m .

2+4+2+3+3=14=2 \times 7 .

\begin{aligned} L(G)= & \left\{\left\{x_{2}, x_{5}\right\},\left\{x_{1}, x_{3}, x_{4}, x_{5}\right\},\left\{x_{2}, x_{4}\right\},\right. \\ & \left.\left\{x_{2}, x_{3}, x_{5}\right\},\left\{x_{1}, x_{2}, x_{4}\right\}\right\} . \end{aligned}

A(G)=\left(\begin{array}{lllll} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 \end{array}\right) .

I(G)=\left(\begin{array}{lllllll} 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right) .

\begin{aligned} & J(G)=\left\{\left\{x_{1}, x_{2}\right\},\left\{x_{2}, x_{3}\right\},\left\{x_{3}, x_{4}\right\},\right. \\ & \left.\left\{x_{4}, x_{5}\right\},\left\{x_{1}, x_{5}\right\},\left\{x_{2}, x_{5}\right\},\left\{x_{2}, x_{4}\right\}\right\} . \end{aligned}

L(G)=\{\emptyset,\{2\},\{2,4\},\{1,1,3\}\} ;

A(G)=\left(\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 2 & 0 & 1 & 0 \end{array}\right) ;

I(G)=\left(\begin{array}{rrrrrr} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & l & 0 & 0 \\ 0 & 0 & -1 & 0 & 1 & -1 \\ -1 & -1 & 0 & 0 & -1 & 1 \end{array}\right) ;

J(G)=\{(4,1),(4,1),(3,2),(2,2),(4,3),(3,4)\} .

m=\binom{n}{2}=\frac{n(n-1)}{2} .

\sigma=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ a & b & c & d & e & f \end{array}\right) .

\alpha(G)+\tau(G)=|X| .

\tau(G) \geq v(G) .

X=N_{0} \cup N_{1} \cup N_{2} \cup N_{3} \cup \ldots \cup N_{k} .

N(S)=\cup_{x \in S} N(x) \backslash S .

\left|N_{G}(S)\right| \geq|S| .

|S|<\left|N_{G}(S)\right| .

\left|S^{\prime}\right| \leq\left|N_{G}\left(S^{\prime}\right)\right| .

\left|S_{0}\right|=\left|N_{G}\left(S_{0}\right)\right| .

A=S_{0} \cup N_{G}\left(S_{0}\right) \text { and } B=(X \cup Y) \backslash A

N_{G_{B}}(S)=N_{G}(S) \backslash N_{G}\left(S_{0}\right),

\left|S_{0}\right|+|S|=\left|S_{0} \cup S\right| \leq\left|N_{G}\left(S_{0} \cup S\right)\right|=\left|N_{G}\left(S_{0}\right)\right|+\left|N_{G_{B}}(S)\right| .

\tau(G)=\mathcal{V}(G) .

M(G)=\max _{X^{\prime} \subseteq X} \min _{x \in G^{\prime}} d(x) .

M(G)-(\omega(G)-1) .

X=N_{0} \cup N_{1} \cup N_{2} \cup \cdots \cup N_{k-1} \cup N_{k}

n-m+f=2 .

m(G) \leq 3 n(G)-6 .

R(G)=\left(r_{1}, r_{2}, \ldots, r_{n}\right)

R(G)=\left(0,0, \ldots, 0, r_{\chi}, r_{\chi+1}, \ldots, r_{n}\right) .

R(G)=(0,1,1) .

P\left(K_{n}, \lambda\right)=\lambda(\lambda-1)(\lambda-2) \cdots(\lambda-n+1) .

P(G, \lambda)=P\left(G_{1}, \lambda\right)+P\left(G_{2}, \lambda\right) .

P(G, \lambda)=P\left(K_{i_{1}}, \lambda\right)+P\left(K_{i_{2}}, \lambda\right)+\cdots+P\left(K_{i_{s}}, \lambda\right)

\begin{gathered} P(G, \lambda)=P\left(K_{i_{1}}, \lambda\right)+P\left(K_{i_{2}}, \lambda\right)+\cdots+P\left(K_{i_{s}}, \lambda\right)= \\ r_{1}(G) P\left(K_{1}, \lambda\right)+r_{2}(G) P\left(K_{2}, \lambda\right)+\cdots+r_{n}(G) P\left(K_{n}, \lambda\right)= \end{gathered}

\sum_{i=1}^{n} r_{i}(G) P\left(K_{i}, \lambda\right) .

P(G, \lambda)=\sum_{i=\chi}^{n} r_{i}(G) \lambda^{(i)}

\begin{gathered} P(G, \lambda)=P\left(K_{3}, \lambda\right)+P\left(K_{2}, \lambda\right)=\lambda^{(3)}+\lambda^{(2)}= \\ \lambda(\lambda-1)(\lambda-2)+\lambda(\lambda-1)=\lambda(\lambda-1)^{2} \end{gathered}

\begin{gathered} P\left(C_{4}, \lambda\right)=0 \cdot P\left(K_{1}, \lambda\right)+1 \cdot P\left(K_{2}, \lambda\right)+2 \cdot P\left(K_{3}, \lambda\right)+1 \cdot P\left(K_{4}, \lambda\right)= \\ \lambda^{(2)}+2 \lambda^{(3)}+\lambda^{(4)}=\lambda(\lambda-1)+2 \lambda(\lambda-1)(\lambda-2)+\lambda(\lambda-1)(\lambda-2)(\lambda-3)= \\ \lambda(\lambda-1)[1+2(\lambda-2)+(\lambda-2)(\lambda-3)]=\lambda(\lambda-1)\left(\lambda^{2}-3 \lambda+3\right)= \\ \lambda^{4}-4 \lambda^{3}+6 \lambda^{2}-3 \lambda . \end{gathered}

R\left(C_{4}\right)=(0,1,2,1) .

P(G, \lambda)=P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right) \cdots P\left(G_{k}, \lambda\right) .

P\left(E_{n}, \lambda\right)=\lambda \cdot \lambda \cdot \ldots \cdot \lambda=\lambda^{n} .

\lambda^{n}=S(n, 1) \lambda^{(1)}+S(n, 2) \lambda^{(2)}+\ldots+S(n, n) \lambda^{(n)}

R\left(E_{n}\right)=(S(n, 1), S(n, 2), \ldots, S(n, n)) .

\lambda^{(n)}=s(n, 1) \lambda+s(n, 2) \lambda^{2}+\ldots+s(n, n) \lambda^{n}

P(G, \lambda)=P\left(G_{1}, \lambda\right)+P\left(G_{2}, \lambda\right)

P\left(G_{1}, \lambda\right)=P(G, \lambda)-P\left(G_{2}, \lambda\right) .

P(G-e, \lambda)=\sum_{i=0}^{n}(-1)^{i} a_{i} \lambda^{n-i}

P(G \cdot e, \lambda)=\Sigma_{i=0}^{n-1}(-1)^{i} b_{i} \lambda^{n-1-i}

\begin{gathered} P(G, \lambda)=P(G-e, \lambda)-P(G \cdot e, \lambda) \\ =\lambda^{n}-(m-1) \lambda^{n-1}+a_{2} \lambda^{n-2}-\cdots+(-1)^{i} a_{i} \lambda^{n-i} \cdots \\ -\left(\lambda^{n-1}-b_{1} \lambda^{n-2}+b_{2} \lambda^{n-2}-\cdots+(-1)^{i} b_{i-1} \lambda^{n-i} \cdots\right) \\ =\lambda^{n}-m(G) \lambda^{n-1}+\left(a_{2}+b_{1}\right) \lambda^{n-2} \cdots(-1)^{i}\left(a_{i}+b_{i-1}\right) \lambda^{n-i} \cdots \end{gathered}

P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right)}{\lambda^{(k)}} .

\begin{gathered} P(G, \lambda)=\frac{P\left(G_{X_{1} \cup S}, \lambda\right)}{\lambda^{(k)}} \frac{P\left(G_{X_{2} \cup S}, \lambda\right)}{\lambda^{(k)}} \lambda^{(k)}= \\ \frac{P\left(G_{X_{1} \cup S}, \lambda\right) P\left(G_{X_{2} \cup S}, \lambda\right)}{\lambda^{(k)}} . \end{gathered}

P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right) \cdots P\left(G_{l}, \lambda\right)}{\left[\lambda^{(k)}\right]^{l-1}} .

\begin{gathered} P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right)}{\lambda(2)}= \\ \frac{\left[\lambda(\lambda-1)\left(\lambda^{2}-3 \lambda+3\right)\right][\lambda(\lambda-1)(\lambda-2)]}{\lambda(\lambda-1)}= \\ \lambda(\lambda-1)(\lambda-2)\left(\lambda^{2}-3 \lambda+3\right) \end{gathered}

P(G, \lambda)=(\lambda-k) P(G-x, \lambda) .

P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(K_{k+1}, \lambda\right)}{\lambda^{(k)}}=

\frac{P(G-x, \lambda) \lambda^{(k+1)}}{\lambda^{(k)}}=(\lambda-k) P(G-x, \lambda) .

P(G, \lambda)=(\lambda-2) P(G-x, \lambda)=\lambda(\lambda-1)(\lambda-2)\left(\lambda^{2}-3 \lambda+3\right) .

r_{i}(G)=(i-k) r_{i}(G-x)+r_{i-1}(G-x) .

S(n, i)=i S(n-1, i)+S(n-1, i-1) .

S(5,3)=25=3 \cdot S(4,3)+S(4,2)=3 \cdot 6+7=25 .

\begin{gathered} P\left(G_{1}, \lambda\right)=\left(\lambda-k_{1}\right) P\left(G_{2}, \lambda\right)=\left(\lambda-k_{1}\right)\left(\lambda-k_{2}\right) P\left(G_{3}, \lambda\right) \cdots= \\ \left(\lambda-k_{1}\right)\left(\lambda-k_{2}\right)\left(\lambda-k_{3}\right) \cdots\left(\lambda-k_{n}\right) \end{gathered}

P(G, \lambda)=\lambda^{S_{0}}(\lambda-1)^{S_{1}}(\lambda-2)^{S_{2}} \cdots(\lambda-\chi(G)+1)^{S_{\chi-1}}

\begin{gathered} P\left(G_{1}, \lambda\right)=(\lambda-2)(\lambda-2)(\lambda-3)(\lambda-2)(\lambda-2)(\lambda-2)(\lambda-1)(\lambda-0)= \\ \lambda(\lambda-1)(\lambda-2)^{5}(\lambda-3) . \end{gathered}

P\left(G_{1}, 4\right)=4(4-1)(4-2)^{5}(4-3)=384 .

P\left(T_{n}, \lambda\right)=\lambda(\lambda-1)^{n-1}

P\left(C_{n}, \lambda\right)=(\lambda-1)^{n}+(-1)^{n}(\lambda-1) .

P\left(C_{n}, \lambda\right)=P\left(T_{n}, \lambda\right)-P\left(C_{n-1}, \lambda\right)=

\begin{gathered} \lambda(\lambda-1)^{n-1}-\left[(\lambda-1)^{n-1}+(-1)^{n-1}(\lambda-1)\right]= \\ (\lambda-1)^{n}+(-1)^{n}(\lambda-1) \end{gathered}

P\left(G^{\prime}, \lambda\right)=\lambda^{s_{0}^{\prime}}(\lambda-1)^{s_{1}^{\prime}} \ldots\left(\lambda-\chi^{\prime}+1\right)^{s_{\chi^{\prime}-1}^{\prime}}

G_{X_{1} \cup A}=G_{1}, G_{X_{2} \cup A}=G_{2}, \ldots, G_{X_{k} \cup A}=G_{k}, G_{A}=G_{0} .

P\left(G^{\prime}, \lambda\right)=\frac{P\left(G_{1}^{\prime}, \lambda\right) P\left(G_{2}^{\prime}, \lambda\right) \ldots P\left(G_{k}^{\prime}, \lambda\right)}{P\left(G_{0}^{\prime}, \lambda\right)^{k-1}} .

P\left(G_{1}, \lambda\right)=(\lambda-p) P\left(G_{1}-x_{0}, \lambda\right) .

P(G, \lambda)=(\lambda-p) P\left(G-x_{0}, \lambda\right) .

P\left(G-x_{0}, \lambda\right)=\frac{P\left(G_{1}-x_{0}, \lambda\right) P\left(G_{2}, \lambda\right) \ldots P\left(G_{k}, \lambda\right)}{P\left(G_{0}, \lambda\right)^{k-1}} .

P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right) \ldots P\left(G_{k}, \lambda\right)}{P\left(G_{0}, \lambda\right)^{k-1}} .

\left\{x_{0}\right\} \cup N\left(x_{0}\right) \subseteq X_{1} \cup A, N\left(x_{0}\right) \cap A \subseteq X_{i} \cup A, 1 \leq i \leq k .

P(G, \lambda)=(\lambda-p) P\left(G-x_{0}, \lambda\right) .

P\left(G-x_{0}, \lambda\right)=\frac{P\left(G_{1}^{\prime}, \lambda\right) P\left(G_{2}^{\prime}, \lambda\right) \ldots P\left(G_{k}^{\prime}, \lambda\right)}{P\left(G_{0}^{\prime}, \lambda\right)^{k-1}},

P\left(G_{1}, \lambda\right)=(\lambda-p) P\left(G_{1}^{\prime}, \lambda\right), \text { and } P\left(G_{i}, \lambda\right)=\left(\lambda-p_{1}\right) P\left(G_{i}^{\prime}, \lambda\right)

P(G, \lambda)=\frac{P\left(G_{1}, \lambda\right) P\left(G_{2}, \lambda\right) \ldots P\left(G_{k}, \lambda\right)}{P\left(G_{0}, \lambda\right)^{k-1}} .

P\left(C_{k}, \lambda\right)=\lambda(\lambda-1)^{2} \lambda(\lambda-1)^{k-2} \lambda^{-2}=(\lambda-1)^{k} \neq P\left(C_{k}, \lambda\right),

W(G, \lambda)=\frac{\lambda P(G, \lambda-1)}{P(G, \lambda)} .

P\left(G^{\prime}, \lambda\right)=P\left(G^{\prime}-x, \lambda\right) W\left(G_{N(x)}^{\prime}, \lambda\right) .

\begin{gathered} P(G, \lambda)=\lambda P(G-x, \lambda-1)=\frac{P(G-x, \lambda)}{P(G-x, \lambda)} \lambda P(G-x, \lambda-1)= \\ P(G-x, \lambda) W\left(G_{N(x)}, \lambda\right) \end{gathered}

P(G, \lambda)=\frac{P(G-x, \lambda) P\left(G_{\{x\} \cup N(x)}, \lambda\right)}{P\left(G_{N(x)}, \lambda\right)}=

P(G-x, \lambda) \frac{\lambda P\left(G_{N(x)}, \lambda-1\right)}{P\left(G_{N(x)}, \lambda\right)}=P(G-x, \lambda) W\left(G_{N(x)}, \lambda\right) .

P(C, \lambda)=P(C-x, \lambda) W\left(C_{N(x)}, \lambda\right)=(\lambda-1)^{k} \neq P(C, \lambda),

P(G, \lambda)=(\lambda-k) P(G-x, \lambda) .

W\left(G_{N(x)}, \lambda\right)=W\left(K_{k}, \lambda\right)=\frac{\lambda(\lambda-1)^{(k)}}{\lambda^{(k)}}=(\lambda-k) .

P(G, \lambda)=P(G-x, \lambda) W\left(G_{N(x)}, \lambda\right)=(\lambda-k) P(G-x, \lambda) .

W\left(G_{N(4)}, \lambda\right)=\frac{\lambda P\left(T_{3}, \lambda-1\right)}{P\left(T_{3}, \lambda\right)}=\frac{\lambda(\lambda-1)(\lambda-2)^{2}}{\lambda(\lambda-1)^{2}}=\frac{(\lambda-2)^{2}}{\lambda-1} .

P\left(G_{1}, \lambda\right)=\lambda(\lambda-1)^{2}(\lambda-2)^{3}(\lambda-3) .

\begin{gathered} P\left(G_{1}, \lambda\right) W\left(G_{N(4)}, \lambda\right)=\lambda(\lambda-1)^{2}(\lambda-2)^{3}(\lambda-3) \frac{(\lambda-2)^{2}}{\lambda-1}= \\ \lambda(\lambda-1)(\lambda-2)^{5}(\lambda-3)=P(G, \lambda) \end{gathered}

M(G)=\max _{X^{\prime} \subseteq X} \min _{x \in G^{\prime}} d(x)

\chi(G) \geq \omega(G) .

\chi^{\prime}(G)=\Delta(G), \text { or } \chi^{\prime}(G)=\Delta(G)+1 \text {. }

\bar{\chi}^{\prime}\left(G_{i}\right)=1, i=1, \ldots, \bar{\chi}^{\prime}(G) .

\bar{\chi}^{\prime}(G)=c+m-n+p .

n-v^{\prime}-p+r .

m^{\prime}=v^{\prime}+\left(n-v^{\prime}-p+r\right)=n-p+r .

\bar{\chi}^{\prime}(G)=c^{\prime}+r+\left(m-m^{\prime}\right)=c^{\prime}+m-n+p,

\bar{\chi}^{\prime}(G)=c^{\prime}+m-n+p \leq c+m-n+p .

\bar{\chi}^{\prime}\left(A_{1}\right) \geq \bar{\chi}^{\prime}\left(A_{0}\right)+1 \geq c+m\left(A_{1}\right)-n\left(A_{1}\right)+p\left(A_{1}\right),

\bar{\chi}^{\prime}(G)=c+m-n+p=1+5-4+1=3,

\bar{\chi}^{\prime}(G)=p-1 .

d(x)+d(y) \geq n,

d\left(x_{1}\right)+d\left(x_{n}\right) \leq(n-2)+(n-2)-(n-3)=n-1

x_{1}, x_{i+1}, x_{i+2}, \ldots, x_{n}, x_{i}, x_{i-1}, \ldots, x_{1}

u \rightarrow x_{4} \rightarrow x_{3} \rightarrow x_{1} \rightarrow x_{2} \rightarrow v

u \rightarrow x_{1} \rightarrow x_{2} \rightarrow v .

u \rightarrow x_{4} \rightarrow x_{3} \rightarrow v .

u \rightarrow x_{4} \rightarrow x_{3} \rightarrow x_{1} \rightarrow x_{2} \rightarrow v .

\Delta(\mathcal{H})=\max _{x \in X}|\mathcal{D}(x)| .

r(\mathcal{H})=\max _{D \in \mathcal{D}}|D| .

(i, j) \text {-entry }= \begin{cases}1 & \text { if } x_{i} \in D_{j}, \\ 0 & \text { if } x_{i} \notin D_{j} .\end{cases}

\begin{gathered} \mathcal{Z}=\left\{X_{1}, X_{2}, \ldots, X_{n}\right\}, \\ X_{i}=\left\{d_{j}: x_{i} \in D_{j} \text { in } \mathcal{H}\right\} . \end{gathered}

\begin{gathered} \mathcal{H}=(X, \mathcal{D}), X=\{1,2,3,4,5,6\}, \mathcal{D}=\left\{D_{1}, D_{2}, D_{3}, D_{4}, D_{5}\right\}, \\ D_{1}=\{1\}, D_{2}=\{1,2\}, D_{3}=\{1,2,4\} \\ D_{4}=\{2,3,5\}, D_{5}=\{3,4,5\} . \end{gathered}

\sum_{i=1}^{n}\left|\mathcal{D}\left(x_{i}\right)\right|=\sum_{j=1}^{m}\left|D_{j}\right| .

3+3+2+2+2+0=12=1+2+3+3+3 .

L=\{\{1\},\{1,2\},\{1,2,4\},\{2,3,5\},\{3,4,5\}\} .

a_{i j}= \begin{cases}1 & \text { if } \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \emptyset \\ 0 & \text { otherwise. }\end{cases}

\left|\mathcal{D}\left(K_{n}^{r}\right)\right|=\binom{n}{r}=\frac{n!}{r!(n-r)!}=\frac{n(n-1)(n-2) \cdots(n-r+1)}{r!} .

\left|\mathcal{D}\left(K_{4}^{3}\right)\right|=\binom{4}{3}=\frac{4 \cdot 3 \cdot 2}{1 \cdot 2 \cdot 3}=4 .

\mu=x_{0} D_{0} x_{1} D_{1} x_{2} \ldots x_{t-1} D_{t-1} x_{t}

\alpha(\mathcal{H})+\tau(\mathcal{H})=|X| .

\tau(\mathcal{H}) \geq v(\mathcal{H}) .

\tau(\mathcal{H})=v(\mathcal{H}) .

\rho(\mathcal{H})=\tau\left(\mathcal{H}^{*}\right) .

L(\mathcal{H})=(\mathcal{D}, \mathcal{E}), \text { where }\left(D_{i}, D_{j}\right) \in \mathcal{E} \Leftrightarrow D_{i} \cap D_{j} \neq \emptyset .

(\mathcal{H})_{2}=(X, \mathcal{E}) \text { where }\left\{x_{i}, x_{j}\right\} \in \mathcal{E} \Leftrightarrow \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \mathbb{O} \text {. }

(\mathcal{H})_{2}=L\left(\mathcal{H}^{*}\right) .

L(\mathcal{H})=\left(\mathcal{H}^{*}\right)_{2} .

\nu(\mathcal{H})=\alpha(L(\mathcal{H})) \text { and } \nu\left(\mathcal{H}^{*}\right)=\alpha\left((\mathcal{H})_{2}\right) .

\left(D_{1} \cap D_{2}\right) \cup\left(D_{1} \cap D_{3}\right) \cup\left(D_{2} \cap D_{3}\right) \subseteq D \text {. }

\left(D_{1} \cap D_{2}\right) \cup\left(D_{1} \cap D_{3}\right) \cup\left(D_{2} \cap D_{3}\right) \subseteq D \text {. }

Y=\left(Y_{1} \cap Y_{2}\right) \cup\left(Y_{1} \cap Y_{3}\right) \cup\left(Y_{2} \cap Y_{3}\right) .

Y \subseteq\left(D_{1} \cap D_{2}\right) \cup\left(D_{1} \cap D_{3}\right) \cup\left(D_{2} \cap D_{3}\right) .

\tau(\mathcal{H})=\nu(\mathcal{H}) .

w(T) \leq \sum_{D \in \mathcal{D}}(|D|-1) .

\wedge(\mathcal{H}, T)=\sum_{D \in \mathcal{D}}(|D|-1)-w(T)+l(\mathcal{H}, T) .

\begin{gathered} \wedge(\mathcal{H}, T)=\sum_{|D|=2}(|D|-1)-w(T)+l(\mathcal{H}, T) \\ =|\mathcal{D}|-(|X|-1)=m-n+1 . \end{gathered}

\begin{gathered} \Lambda(\mathcal{H}, T)=\sum_{D \in \mathcal{D}}(|D|-1)-w(T)+l(\mathcal{H}, T) \\ =2+2+0+0-3+1=2 . \end{gathered}

\wedge(\mathcal{H}, T)=\sum_{D \in \mathcal{D}}(c(T, D)-1)+l(\mathcal{H}, T) .

c(T, D)=|D|-\left|E_{D}\right| .

w(T)=\sum_{|D| \geq 2}\left|E_{D}\right| .

\begin{gathered} \wedge(\mathcal{H}, T)=\sum_{D \in \mathcal{D}}(|D|-1)-w(T)+l(\mathcal{H}, T) \\ =\{\text { ignore the singletons }\}=\sum_{|D| \geq 2}(|D|-1)-w(T)+l(\mathcal{H}, T) \\ =\{\text { insert the weight of the forest }\} \\ =\sum_{|D| \geq 2}(|D|-1)-\sum_{|D| \geq 2}\left|E_{D}\right|+l(\mathcal{H}, T) \\ =\{\text { unite the sum }\}=\sum_{|D| \geq 2}\left(|D|-\left|E_{D}\right|-1\right)+l(\mathcal{H}, T) \\ =\{\text { use (8.5) }\}=\sum_{|D| \geq 2}(c(T, D)-1)+l(\mathcal{H}, T) \\ =\{\text { bring in singletons back }\}=\sum_{D \in \mathcal{D}}(c(T, D)-1)+l(\mathcal{H}, T) . \end{gathered}

\wedge(\mathcal{H}, T)=(2-1)+(1-1)+(1-1)+1=2 .

w(T)=\sum_{D \in \mathcal{D}}(|D|-1) .

\wedge(\mathcal{H}, T)=\wedge\left(\mathcal{H}_{1}, T_{1}\right)+|\mathcal{D}(x)|-\left|\mathcal{D}_{T}(x)\right|-m_{2} .

\wedge(\mathcal{H}, T)=\wedge\left(\mathcal{H}_{1}, T_{1}\right) \text { if and only if }|\mathcal{D}(x)|=\left|\mathcal{D}_{T}(x)\right|+m_{2} .

\wedge(\mathcal{H}, T)=2+2+1+0+0-3+1=3 .

\wedge(\mathcal{H}-x, T-x)=2+1+0+0-2+1=2 .

\wedge(\mathcal{H}-x, T-x)+|\mathcal{D}(x)|-\left|\mathcal{D}_{T}(x)\right|-m_{2}=2+3-1-1=3=\wedge(\mathcal{H}, T) .

w\left(T_{1}\right)=w(T)-\left|\mathcal{D}_{T}(x)\right| .

w\left(T^{\prime}\right)=w\left(T_{1}\right)+\left|\mathcal{D}_{T}(x)\right| \geq w\left(T_{1}\right)+1+\left|\mathcal{D}_{T}(x)\right|=w(T)+1

w(\mathcal{H})=\max _{T} w(T) .

\wedge(\mathcal{H})=\min _{T} \wedge(\mathcal{H}, T)=\sum_{D \in \mathcal{D}}(|D|-1)-w(\mathcal{H}) .

\wedge(\mathcal{H})=\wedge\left(\mathcal{H}_{1}\right) .

\wedge(\mathcal{H})=\wedge(\mathcal{H}, T)=\wedge\left(\mathcal{H}_{1}, T_{1}\right) .

\mu=x_{0} D_{0} x_{1} D_{1} x_{2} \ldots x_{t-1} D_{t-1} x_{t}

\chi^{\prime}(\mathcal{H}) \geq \omega(L(\mathcal{H})) \geq \Delta(\mathcal{H}) .

\chi^{\prime}\left(\mathcal{H}^{\prime}\right)=\Delta\left(\mathcal{H}^{\prime}\right) .

\chi^{\prime}(\mathcal{H})=\chi(L(\mathcal{H}))=\omega(L(\mathcal{H}))=\Delta(\mathcal{H}) .

n-\sum_{i=1}^{m}\left(\left|E_{i}\right|-1\right)+f=m-\sum_{j=1}^{n}\left(\left|\mathcal{D}\left(x_{j}\right)\right|-1\right)+f=2 .

m^{\prime}=\sum_{j=1}^{m}\left|D_{j}\right|=\sum_{i=1}^{n}\left|\mathcal{D}\left(x_{i}\right)\right|

n^{\prime}-m^{\prime}+f^{\prime}=2

n-\sum_{i=1}^{m}\left(\left|D_{i}\right|-1\right)+f=2

m-\sum_{j=1}^{n}\left(\left|\mathcal{D}\left(x_{j}\right)\right|-1\right)+f=2

n-\sum_{i=1}^{m}\left(\left|E_{i}\right|-1\right)+f=4-(2+2)+2=2

\left.m-\sum_{j=1}^{n}\left(\left|\mathcal{D}\left(x_{j}\right)\right|-1\right)+f=2-(0+1+0+1)+2\right)=2 .

X=\cup_{i=1}^{k} S_{i}, \quad S_{i} \neq \emptyset, \quad S_{i} \cap S_{j}=\emptyset, i \neq j .

\begin{gathered} \alpha(\mathcal{H}) \chi(\mathcal{H}) \geq n ; \\ \chi(\mathcal{H}) \leq \tau(\mathcal{H})+1=n-\alpha(\mathcal{H})+1 . \end{gathered}

\left\lfloor\frac{|D|}{\lambda}\right\rfloor \leq\left|D \cap S_{i}\right| \leq\left\lceil\frac{|D|}{\lambda}\right\rceil,

\min \{|D|, \lambda\}

\lambda \leq \min _{D \in \mathcal{D}}|D|,

\left\lfloor\frac{n}{\lambda}\right\rfloor \leq\left|S_{i}\right| \leq\left\lceil\frac{n}{\lambda}\right\rceil .

0 \leq a_{j} \leq b_{j} \leq\left|D_{j}\right| .

a_{j} \leq\left|D_{j} \cap S_{i}\right| \leq b_{j} .

\chi(\mathcal{H}) \leq \wedge(\mathcal{H})+2 .

D_{i}, D_{j} \in \mathcal{D}_{1}(x) \Rightarrow D_{i} \cap D_{j}=\{x\} .

M(\mathcal{H})=\max _{Y \subseteq X} \min _{x \in Y} m\left(x, \mathcal{H}_{Y}\right) .

m\left(y, \mathcal{H}_{Y}\right)=\min _{z} m\left(z, \mathcal{H}_{Y}\right)=M(\mathcal{H}) .

M(\mathcal{H})=m\left(y, \mathcal{H}_{Y}\right) \leq m\left(x_{k}, \mathcal{H}_{k}\right) \leq t .

\chi(\mathcal{H}) \leq M(\mathcal{H})+1 .

m(1, \mathcal{H})=2, m(2, \mathcal{H})=2, m(3, \mathcal{H})=2, m(4, \mathcal{H})=1, m(5, \mathcal{H})=1 .

5,4,1,3,2 .

2,3,1,4,5,

c=(2,1,2,1,1) .

c: X \rightarrow\{1,2, \ldots, \lambda\} .

R(\mathcal{H})=\left(r_{1}, r_{2}, \ldots, r_{n}\right)

R(\mathcal{H})=\left(0, \ldots, 0, r_{\chi}, \ldots, r_{\bar{\chi}}, 0, \ldots, 0\right) .

P(\mathcal{H}, \lambda)=\sum_{i=\chi(\mathcal{H})}^{\bar{\chi}(\mathcal{H})} r_{i}(\mathcal{H}) \lambda^{(i)} .

r_{\bar{\chi}} \lambda(\lambda-1)(\lambda-2) \cdots(\lambda-\bar{\chi}+1)

\chi(\mathcal{H})=\bar{\chi}(\mathcal{H})=0, \quad R(\mathcal{H})=(0,0, \ldots, 0), \quad P(\mathcal{H}, \lambda)=0 .

1 \leq \chi\left(\mathcal{H}_{\mathcal{D}}\right) \leq \chi(\mathcal{H}) \leq \bar{\chi}(\mathcal{H}) \leq \bar{\chi}\left(\mathcal{H}_{C}\right) \leq n .

1 \leq \chi\left(\mathcal{H}_{\mathcal{D}}\right)=2 \leq \chi(\mathcal{H})=2 \leq \bar{\chi}(\mathcal{H})=4 \leq \bar{\chi}\left(\mathcal{H}_{\mathcal{C}}\right)=4 \leq n=5 .

b(x, \mathcal{H})=\max _{y}\{|\mathcal{C}(x) \cap \mathcal{C}(y)|: y \in X, y \neq x\}

o(x, \mathcal{H})=|\mathcal{C}(x)|-b(x, \mathcal{H}) \geq 0

O(\mathcal{H})=\max _{Y \subseteq X} \min _{x \in Y} o\left(x, \mathcal{H}_{Y}\right)

o\left(y, \mathcal{H}^{\prime}\right)=\min _{z} o\left(z, \mathcal{H}^{\prime}\right)=O(\mathcal{H}) \geq t+1 .

o\left(x_{i}, \mathcal{H}_{i},\right) \leq O(\mathcal{H})=\max _{Y \subseteq X} \min _{x \in Y} o(x, \mathcal{H} / Y,)

P(\mathcal{H}, \lambda)=P\left(\mathcal{H}_{1}, \lambda\right)+P\left(\mathcal{H}_{2}, \lambda\right), \quad R(\mathcal{H})=R\left(\mathcal{H}_{1}\right)+R\left(\mathcal{H}_{2}\right),

\begin{gathered} \mathcal{H}_{1}=\left(X, \mathcal{C}, \mathcal{D}_{1}\right), \mathcal{D}_{1}=\mathcal{D} \cup\left\{x_{k}, x_{l}\right\}, \\ \mathcal{H}_{2}=\left(X, \mathcal{C}_{1}, \mathcal{D}\right), \mathcal{C}_{1}=\mathcal{C} \cup\left\{x_{k}, x_{l}\right\} \text { (splitting). } \end{gathered}

P(\mathcal{H}, \lambda)=P\left(\mathcal{H}_{1}, \lambda\right), \quad R(\mathcal{H})=R\left(\mathcal{H}_{1}\right), \text { where }

\mathcal{C}_{1}=\mathcal{C}-C_{t} \text { (contraction). }

\begin{gathered} Z=\left\{K_{3}, K_{3}, K_{2}\right\}, R(\mathcal{H})=(0,1,2,0), \\ P(\mathcal{H}, \lambda)=2 \lambda^{(3)}+\lambda^{(2)}=2 \lambda^{3}-5 \lambda^{2}+3 \lambda, \end{gathered}

L=\{(1,2,3,2),(1,2,1,3),(1,2,1,2)\} .

R(\mathcal{H})=R(\mathcal{H}-C)(R(\mathcal{H})=R(\mathcal{H}-D)) .

R(\mathcal{H})=R\left(\mathcal{H}-C_{j}\right)=(1,7,0,0), j=1,2,3,4 .

\mathcal{D}=\{\{1,2\},\{2,3\},\{3,4\},\{4,1\}\}

\begin{gathered} \chi(\mathcal{H})=\max \left\{\chi\left(\mathcal{H}_{1}\right), \chi\left(\mathcal{H}_{2}\right), \ldots, \chi\left(\mathcal{H}_{k}\right)\right\}, \\ \bar{\chi}(\mathcal{H})=\bar{\chi}\left(\mathcal{H}_{1}\right)+\bar{\chi}\left(\mathcal{H}_{2}\right)+\ldots+\bar{\chi}\left(\mathcal{H}_{k}\right), \\ P(\mathcal{H}, \lambda)=P\left(\mathcal{H}_{1}, \lambda\right) P\left(\mathcal{H}_{2}, \lambda\right) \ldots P\left(\mathcal{H}_{k}, \lambda\right) . \end{gathered}

v(k)=k+4 .

\mathcal{K}(n, l, m)=(X, \mathcal{C}, \mathcal{D})=\left(X,\binom{X}{l},\binom{X}{m}\right) .

n \geq(l-1)(m-1)+1 .

(l-1)(m-1) \leq n .

N_{n}=\sum_{k=1}^{n}\left\lfloor\frac{n}{k}\right\rfloor \simeq n \log n

\lim _{n \rightarrow \infty} \frac{N_{n}}{n^{2}}=\lim _{n \rightarrow \infty} \frac{\log n}{n}=0,

\chi(\mathcal{H})=\bar{\chi}(\mathcal{H})=\chi, r_{\chi}(\mathcal{H})=1,

R(\mathcal{H})=(0, \ldots, 0,1,0, \ldots, 0) .

P(\mathcal{H}, \lambda)=\lambda(\lambda-1)(\lambda-2) \cdots(\lambda-\chi+1)=\lambda^{(\chi)} .

x_{j} \neq x_{j+1} \neq x_{j+2} \neq x_{j} \Rightarrow\left\{x_{j}, x_{j+1}, x_{j+2}\right\} \in \mathcal{C}, j=0,1, \ldots, k-2 .

R\left(\mathcal{H}_{1}\right)=R\left(\mathcal{H}_{2}\right)=(0,1,0,0), P\left(\mathcal{H}_{1}, \lambda\right)=P\left(\mathcal{H}_{2}, \lambda\right)=\lambda(\lambda-1) .

|C| \geq n-\chi .

L_{1}=\left(X_{1}, E_{1}\right), L_{2}=\left(X_{2}, E_{2}\right), \ldots, L_{\chi}=\left(X_{\chi}, E_{\chi}\right) .

\left|X_{1}\right|-1+\left|X_{2}\right|-1+\ldots+\left|X_{\chi}\right|-1=|X|-\chi .

M=\sum_{\substack{x \in T \\ d(x) \geq 2}}\binom{d(x)}{2},

\left\{x_{1}, x_{2}, \ldots, x_{i}\right\}, i=1,2, \ldots, n

\boldsymbol{\sigma}=\left(x_{5}, x_{3}, x_{4}, x_{2}, x_{1}\right) .

\bar{\chi}(\mathcal{H}) \leq \alpha_{c}(\mathcal{H}) .

\boldsymbol{\alpha}_{c}(\mathcal{H})-\overline{\boldsymbol{\chi}}(\mathcal{H})>k .

C_{i} \cap C_{j}=\{1\}, \quad i, j \in I=\{1,2, \ldots k+2\}, \quad i \neq j .

\bar{\chi}\left(\mathcal{H}^{\prime}\right)=\alpha_{c}\left(\mathcal{H}^{\prime}\right) .

\begin{aligned} & n_{12}+n_{23}+n_{34}+2<r, \\ & n_{12}+n_{14}+n_{34}+2<r . \end{aligned}

\begin{aligned} & n_{12}+1+n_{23}<r, \\ & n_{23}+1+n_{31}<r, \end{aligned}

n_{31}+1+n_{12}<r .

X=\{1,2,3, \ldots, 2 r\},

\mathcal{C}=\left\{C_{1}, C_{2}, \ldots, C_{2 r}, C_{o}, C_{e}\right\}

C_{1}=\{1,2,4,6,8, \ldots, 2 r-2\} .

\bar{\chi}\left(\mathcal{H}^{r}\right)=2 r-4<2 r-3=\alpha_{\mathcal{C}}\left(\mathcal{H}^{r}\right) .

\bigcap_{C \in \mathcal{C}_{1}} C \neq \emptyset .

\bar{\chi}(\mathcal{H})=\alpha_{C}(\mathcal{H})=|X|-\tau\left(\mathcal{H}_{C}\right)=|X|-\nu\left(\mathcal{H}_{C}\right) .

\nu\left(\mathcal{H}_{C}\right)=\tau\left(\mathcal{H}_{C}\right) .

\begin{aligned} & V_{1}=(\{1,2,3,4\},\{\{1,2,3\},\{1,3,4\},\{1,2,4\}\}) \text { (monostar); } \\ & V_{2}=(\{1,2,3,4,5\},\{\{1,2,3\},\{1,4,5\}\}) \text { (monostar); } \\ & V_{3}=(\{1,2,3,4,5\},\{\{1,2,3\},\{1,3,4\},\{1,4,5\}\}) \text { (monostar); } \\ & V_{4}=(\{1,2,3,4,5\},\{\{1,2,3\},\{1,3,4\},\{1,4,5\},\{1,2,5\}\}) \text { (monostar); } \\ & \left.V_{5}=(\{1,2,3,4,5\},\{\{1,2,3\},\{2,3,4\},\{3,4,5\},\{4,5,1\},\{5,1,2\}\}) \text { (cycloid } C_{5}^{3}\right) ; \\ & K_{1}=(\{1,2,3,4,5,6\},\{\{1,2,4\},\{2,3,5\},\{3,4,6\}, \quad\{4,5,1\}, \quad\{5,6, \quad 2\}, \quad\{6,1,3\}, \\ & 3,5\},\{2,4,6\}\}) \text { (Král's example). } \end{aligned}

R(\mathcal{H})=(0,7,0,0)

1122,1112,1121,1222,2122,1212,1221.

R(\mathcal{H})=\left(0, r_{2}, 0,1,0,0,0,0\right) .

\left\{x_{1}, x_{2}, a_{1}, \ldots, a_{t-2}, b_{1}, \ldots, b_{t-2}\right\} .

R\left(\mathcal{H}_{2,4}\right)=(0,4,0,1,0,0),

P(\mathcal{H}, \lambda)=\lambda(\lambda-1)\left(\lambda^{2}-5 \lambda+10\right) .

R(\mathcal{H})=\left(0, \ldots, 0, r_{\chi}, \ldots, r_{\bar{\chi}}, 0, \ldots, 0\right) .

R\left(\mathcal{H}^{\prime}\right)=\left(0, r_{2}\left(\mathcal{H}^{\prime}\right), 0 \ldots, 0, r_{\chi}, \ldots, r_{\bar{\chi}}, 0, \ldots, 0\right) .

(A, A, B, B, C, C, D) .

\{1,2,7\},\{3,4,7\},\{5,6,7\} \in \mathcal{E} .

\begin{array}{lll} 123 & 341 & 561 \\ 124 & 342 & 562 \\ 125 & 345 & 563 \\ 126 & 346 & 564 . \end{array}

r_{k}(G)=\sum_{i \geq 1} S(i, k-1) f_{i}\left(G^{*}\right), 1 \leq k \leq n(G),

P(\mathcal{H}, \lambda)=\sum_{i \geq 1} f_{i}\left(G^{*}\right) \lambda(\lambda-1)^{i} .

\mathcal{H}=\left(X,\binom{X}{3},\binom{X}{3}\right)

X=\{1,2,3,4\}, \mathcal{C}=\mathcal{D}=\{\{1,2,3\},\{2,3,4\},\{3,4,1\},\{4,1,2\}\} ;

R\left(\mathcal{H}_{2,4}^{\prime}\right)=(0,1,0,1,0,0),

P\left(\mathcal{H}_{2,4}^{\prime}, \lambda\right)=\lambda(\lambda-1)\left(\lambda^{2}-5 \lambda+7\right)

f(n) \leq C g(n)

A\left(G_{1}\right)=\left(\begin{array}{llll} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array}\right)

I\left(G_{1}\right)=\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right)