An Extremal Problem On Potentially Kr+1−(kP2⋃tK2)K_{r+1}-(kP_2\bigcup tK_2)-graphic Sequences

A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine $\sigma (K_{r+1}-(kP_2\bigcup tK_2), n)$ for $n\geq 4r+10, r+1 \geq 3k+2t, k+t \geq 2,k \geq 1, t \geq 0$ .Comment: 6 page

An Extremal Problem On Potentially Kr+1−HK_{r+1}-H-graphic Sequences

Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-H, n)$ for $n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ where $H$ is a graph on $k$ vertices which contains a tree on 4 vertices but not contains a cycle on 3 vertices. We also determine the values of $\sigma (K_{r+1}-P_2, n)$ for $n\geq 4r+8, r\geq 3$. There are a number of graphs on $k$ vertices which containing a tree on 4 vertices but not containing a cycle on 3 vertices (for example, the cycle on $k$ vertices, the tree on $k$ vertices, and the complete 2-partite graph on $k$ vertices, etc).Comment: 10 page

On potentially Kr+1−UK_{r+1}-U-graphical Sequences

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} \bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3\bigcup C_{i_1} \bigcup C_{i_2} \bigcup >... \bigcup C_{i_p}$ $(i_j\neq 4, j=2,3,..., p, i_1 \geq 5)$, $C_3\bigcup P_{i_1} \bigcup P_{i_2} \bigcup ... \bigcup P_{i_p}$ $(i_1 \geq 3)$, $C_3\bigcup P_{i_1} \bigcup C_{i_2} \bigcup >... \bigcup C_{i_p}$ $(i_j\neq 4, j=2,3,..., p, i_1 \geq 3)$, etc)Comment: 10 page

The smallest degree sum that yields potentially K_{r+1}-Z-graphical Sequences

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on 4 vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on 4 vertices.Comment: 13 page

On Potentially 3-regular graph graphic Sequences

For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25].Comment: 19 page

On Potentially K_5-E_3-graphic Sequences

Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_3$, $K_5-A_3$, $K_5-K_3$ and $K_5-K_{1,3}$-graphic sequences where $A_3$ is $P_2\cup K_2$. Moreover, we also characterize the potentially $K_5-2K_2$-graphic sequences where $pK_2$ is the matching consisted of $p$ edges.Comment: 25 page

Exact solution to an extremal problem on graphic sequences with a realization containing every 22-tree on kk vertices

A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi=(d_1,\ldots,d_n)$ of nonnegative integers is a {\it graphic sequence} if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795--802) proved that if $k\ge 2$, $n\ge \frac{9}{2}k^2+\frac{19}{2}k$ and $\pi=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n d_i>(k-2)n$, then $\pi$ has a realization containing every 1-tree (the usual tree) on $k$ vertices. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erd\H{o}s and S\'{o}s. In this paper, we investigate an analogue problem for $2$-trees and prove that if $k\ge 3$ is an integer with k\equiv i(\mbox{mod }3), $n\geq20\lfloor\frac{k}{3}\rfloor^2+31\lfloor\frac{k}{3}\rfloor+12$ and $\pi=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n d_i>\max\{(k-1)(n-1),2\lfloor\frac{2k}{3}\rfloor n-2n-\lfloor\frac{2k}{3}\rfloor^2+\lfloor\frac{2k}{3}\rfloor+1-(-1)^i\}$, then $\pi$ has a realization containing every 2-tree on $k$ vertices. Moreover, the lower bound $\max\{(k-1)(n-1),2\lfloor\frac{2k}{3}\rfloor n-2n-\lfloor\frac{2k}{3}\rfloor^2+\lfloor\frac{2k}{3}\rfloor+1-(-1)^i\}$ is the best possible. This result implies a conjecture due to Zeng and Yin (Discrete Math. Theor. Comput. Sci., 17(3)(2016), 315--326).Comment: 31 pag

An extremal problem on potentially Kr+1 − H-graphic sequences, accepted by Ars Combinatoria

A sequence S is potentially Km −H-graphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − H, n) is potentially Km − H-graphical. In this paper, we determined the values of σ(Kr+1 − H, n) for n ≥ 4r + 10, r ≥ 3, r + 1 ≥ k ≥ 4 and H be a graph on k vertices which containing a tree on 4 vertices but not containing a cycle on 3 vertices and σ(Kr+1 − P2, n) for n ≥ 4r + 8, r ≥ 3. Key words: graph; degree sequence; potentially Kr+1 − H-graphic sequence AMS Subject Classifications: 05C07, 05C35

An extremal problem on potentially Kr+1 − H-graphic sequences, accepted by Ars Combinatoria

Let Kk, Ck, Tk, and Pk denote a complete graph on k vertices, a cycle on k vertices, a tree on k + 1 vertices, and a path on k + 1 vertices, respectively. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). A sequence S is potentially Km − H-graphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − H, n) is potentially Km − H-graphical. In this paper, we determine the values of σ(Kr+1−H, n) for n ≥ 4r+10, r ≥ 3, r+1 ≥ k ≥ 4 where H is a graph on k vertices which contains a tree on 4 vertices but not contains a cycle on 3 vertices. We also determine the values of σ(Kr+1 − P2, n) for n ≥ 4r + 8, r ≥ 3. Key words: graph; degree sequence; potentially Kr+1 − H-graphic sequence AMS Subject Classifications: 05C07, 05C35

An extremal problem on potentially Kr+1 − H-graphic sequences, accepted by Ars Combinatoria

Let Kk, Ck, Tk, and Pk denote a complete graph on k vertices, a cycle on k vertices, a tree on k + 1 vertices, and a path on k + 1 vertices, respectively. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). A sequence S is potentially Km − H-graphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − H, n) is potentially Km − H-graphical. In this paper, we determine the values of σ(Kr+1−H, n) for n ≥ 4r+10, r ≥ 3, r+1 ≥ k ≥ 4 where H is a graph on k vertices which contains a tree on 4 vertices but not contains a cycle on 3 vertices. We also determine the values of σ(Kr+1 − P2, n) for n ≥ 4r + 8, r ≥ 3. Key words: graph; degree sequence; potentially Kr+1 − H-graphic sequence AMS Subject Classifications: 05C07, 05C35