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 $σ(K_4-e, 6)= 20$.
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…
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 $S$ 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 $n\geq 4r+10, r+1 \geq 3k+2t, k+t \geq 2,k \geq 1, t \geq 0$ .
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 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 $m=k+2=5$.
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…
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 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_…
In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra.
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 purpose of this note is to exhibit clearly how the "graphical condensation" identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn-Percus method for the enumeration of matchings. Knuth termed the relevant identities "overlapping Pfaffian" identities and the key concept of proof "superpositions of matchings". In our uniform presentation of the material, we also give an apparently unpublished general "overlapping Pfaffian" identity of Krattenthaler. A previous version of this paper contained an erroneous application of the Kasteleyn-Percus method, wh…
One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.
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…
In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,yD_x)$ and $\mathbb{E}(a,b,c,d,e,yθ_x)$ to derive two potentially useful generalizations of the $q$-binomial theorem, a set of two extensions of the $q$-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the $q$-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.
