A sequence $S$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical if it has a realization containing a $K_{p_{1},p_{2},...,p_{t}}$ as a subgraph, where $K_{p_{1},p_{2},...,p_{t}}$ is a complete t-partite graph with partition sizes $p_{1},p_{2},...,p_{t} (p_{1}\geq p_{2}\geq ...\geq p_{t} \geq 1)$. Let $σ(K_{p_{1},p_{2},...,p_{t}}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p_{1},p_{2},...,p_{t}}, n)$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical. In this paper, we prove that $σ(K_{p_{1},p_{2},...,p_{t}}, n)\geq 2[((2p_{1}+2p_{2}+...+2p_{t}-p_{1}-p_{2}-...-p_{i}-2)n -(p_{1}+p_{2}+...+p_{t}-p_{i})(p_{i}+p_{i+1}+...+p_{t}-1)+2)/2]$ for $n \geq p_{1}+p_{2}+...+p_{t}, i=2,3,...,t.$
A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine $σ(K_{r+1}-(kP_2\bigcup tK_2), n)$ for $n\geq 4r+10, r+1 \geq 3k+2t, k+t \geq 2,k \geq 1, t \geq 0$ .
A sequence $S$ is potentially $K_{m}-C_{4}$-graphical if it has a realization containing a $K_{m}-C_{4}$ as a subgraph. Let $σ(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-C_{4}, n)$ is potentially $K_{m}-C_{4}$-graphical. In this paper, we prove that $σ(K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq 4.$ We conjecture that equality holds for $n \geq m \geq 4.$ We prove that this conjecture is true for $m=5$.
A sequence $S$ is potentially $K_{m}-P_{k}$ graphical if it has a realization containing a $K_{m}-P_{k}$ as a subgraph. Let $σ(K_{m}-P_{k}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-P_{k}, n)$ is potentially $K_{m}-P_{k}$ graphical. In this paper, we prove that $σ(K_{m}-P_{k}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq k+1\geq 4.$ We conjecture that equality holds for $n \geq m \geq k+1\geq 4.$ We prove that this conjecture is true for $m=k+1=5$ and $m=k+2=5$.
A sequence $S$ is potentially $K_{p,1,1}$ graphical if it has a realization containing a $K_{p,1,1}$ as a subgraph, where $K_{p,1,1}$ is a complete 3-partite graph with partition sizes $p,1,1$. Let $σ(K_{p,1,1}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p,1,1}, n)$ is potentially $K_{p,1,1}$ graphical. In this paper, we prove that $σ(K_{p,1,1}, n)\geq 2[((p+1)(n-1)+2)/2]$ for $n \geq p+2.$ We conjecture that equality holds for $n \geq 2p+4.$ We prove that this conjecture is true for $p=3$.
A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of gra…
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. In this paper, we characterize the potentially $K_6-C_5$ -graphic sequences.
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ 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].
In this paper, we characterize the potentially $(K_5-C_4)$-graphic sequences where $K_5-C_4$ is the graph obtained from $K_5$ by removing four edges of a 4 cycle $C_4$. This characterization implies a theorem due to Lai [6].
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_6-C_4$-graphic sequences. This characterization implies a theorem due to Hu and Lai [7].
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize potentially $K_{2,5}$-graphic sequences. This characterization implies a special case of a theorem due to Yin et al. [26].
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.
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.
A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $σ(K_4-e, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_4-e, n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $σ(K_4-e, n)$. In this paper, we prove that $σ(K_4-e, n)=2[(3n-1)/2]$ for $n\geq 7$, and $n=4,5,$ and $σ(K_4-e, 6)= 20$.
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 $K_5-P_4$ and $K_5-Y_4$-graphic sequences where $Y_4$ is a tree on 5 vertices and 3 leaves.
A sequence of nonnegative integers $π$ is {\it graphic} if it is the degree sequence of some graph $G$. In this case we say that $G$ is a \textit{realization} of $π$, and we write $π=π(G)$. A graphic sequence $π$ is {\it potentially $H$-graphic} if there is a realization of $π$ that contains $H$ as a subgraph. Given nonincreasing graphic sequences $π_1=(d_1,\ldots,d_n)$ and $π_2 = (s_1,\ldots,s_n)$, we say that $π_1$ {\it majorizes} $π_2$ if $d_i \geq s_i$ for all $i$, $1 \leq i \leq n$. In 1970, Erdős showed that for any $K_{r+1}$-free graph $H$, there exists an $r$-partite graph $G$ such tha…
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we d…
The set of all non-increasing nonnegative integers sequence $π=$ ($d(v_1),$ $d(v_2),$ $...,$ $d(v_n)$) is denoted by $NS_n$. A sequence $π\in NS_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $π$. The set of all graphic sequences in $NS_n$ is denoted by $GS_n$. A graphical sequence $π$ is potentially $H$-graphical if there is a realization of $π$ containing $H$ as a subgraph, while $π$ is forcibly $H$-graphical if every realization of $π$ contains $H$ as a subgraph. Let $K_k$ denote a complete graph o…
A non-increasing sequence $π=(d_1,\ldots,d_n)$ of nonnegative integers is said to be graphic if it is realizable by a simple graph $G$ on $n$ vertices. A graphic sequence $π=(d_1,\ldots,d_n)$ is said to be potentially $_3C_\ell$-graphic if there is a realization of $π$ containing cycles of every length $r$, $3\le r\le \ell$. It is well-known that if the non-increasing degree sequence $(d_1,\ldots,d_\ell)$ of a graph $G$ on $\ell$ vertices satisfies the Pósa condition that $d_{\ell+1-i}\ge i+1$ for every $i$ with $1\le i<\frac{\ell}{2}$, then $G$ is either pancyclic or bipartite. In this paper,…
In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $σ(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $σ(S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $σ(H, n)$. We show $σ(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $σ(C_{2m+2}, n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$.
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices a…
A sequence of nonnegative integers π=(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence π. In this case, G is said to realize or be a realization of π. Degree sequence results in the literature generally fall into two classes: forcible problems, in which all realizations of a graphic sequence must have a given property, and potential problems, in which at least one realization of πmust have the given property. Given a graph H, a graphic sequence πis potentially H-graphic if there is some realization of πthat contains H as a subgraph. In 1991, Erdős, Jacobson and L…
The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular graphic bicyclic sequences. Moreover, we obtain the relations of the second Zagreb indices among the extremal graphs with different degree sequences.
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 $π=(d_1,\ldots,d_n)$ 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 $π=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n d_i>(k-2)n$, then $π$ has a re…
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices a…
As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some "yes" instances but with very low error rate.
A graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. The Erdős-Jacobson-Lehel problem asks to determine $σ(H,n)$, the minimum even integer such that any $n$-term graphic sequence $π$ with sum at least $σ(H,n)$ is potentially $H$-graphic. The parameter $σ(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially $H$-graphic…
A sequence of nonnegative integers $π=(d_1,d_2,...,d_n)$ is graphic if there is a (simple) graph $G$ of order $n$ having degree sequence $π$. In this case, $G$ is said to realize or be a realization of $π$. Given a graph $H$, a graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. In this paper, we consider a degree sequence analogue to classical graph Ramsey numbers. For graphs $H_1$ and $H_2$, the potential-Ramsey number $r_{pot}(H_1,H_2)$ is the minimum integer $N$ such that for any $N$-term graphic sequence $π$, either $π$ is p…
Let $n$ be even, let $π= (d_1, \ldots, d_n)$ be a graphic degree sequence, and let $π- k = (d_1 - k, \ldots, d_n - k)$ also be graphic. Kundu proved that $π$ has a realization $G$ containing a $k$-factor, or $k$-regular graph. Another way to state the conclusion of Kundu's theorem is that $π$ \emph{potentially} contains a $k$-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: $π$ potentially contains $k$ edge-disjoint $1$-factors. Along these lines, they proved $π$ would potentially contain edge-disjoint copies of a $(k-2)$-factor and two $1$-factors. We f…
We consider sufficient conditions for a degree sequence $π$ to be forcibly $k$-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially $k$-factor graphical. We first give a theorem for $π$ to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most $β\ge0$. These theorems are equal in strength to Chvátal's well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for $π$ to be forcibly 2-fa…
This survey is two-fold. We first report new progress on the spectral extremal results on the Turán type problems in graph theory. More precisely, we shall summarize the spectral Turán function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By …
