Related papers: On Potentially $(K_5-H)$-graphic Sequences

Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in $K_{a,b}, 2 \leq a \leq b$…

We show that for all simple graphs G other than the cycles C_3,C_4,C_5, and the claw K_1,3 there exists a K > 0 such that whenever k > K the k-th iterate of the line graph can be distinguished by at most two colors. Additionally we…

Let $G_1\sq G_2$ denote the graph obtained from $G_1+G_2$ by adding new edges from each vertex of $G_1$ to every vertex of $G_2$. In this paper, the critical groups of the graphs $K_m\sq P_n$$(n\geq4)$ and $P_m\sq P_n$$(m\geq4,n\geq5)$ are…

The topological containment problem is known to be polynomial-time solvable for any fixed pattern graph $H$, but good characterisations have been found for only a handful of non-trivial pattern graphs. The complete graph on five vertices,…

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$…

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the…

A graph is \emph{$(\mathcal{I}, \mathcal{F})$-partitionable} if its vertex set can be partitioned into two parts such that one part $\mathcal{I}$ is an independent set, and the other $\mathcal{F}$ induces a forest. A graph is…

In answer to a question of Eggleton, we prove that the complete multigraph on 5 vertices with edge multiplicity 6, namely $K_{5}^{(6)}$, has a decomposition into 5 copies of the family of trees of order 5 and that $K_{7}^{(22)}$ has a…

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we determine the structure of all finite groups $G$ with $K_4$-free character graph…

Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined $k$-leaf powers as the class of graphs $G=(V,E)$ which has a $k$-leaf root $T$, i.e., $T$ is a tree such that the vertices of $G$ are exactly the…

Every $K_4$-free graph on $n$ vertices has a set of $\lfloor n/2\rfloor$ vertices spanning at most $n^2/18$ edges.

A graph $G$ is divisible by a graph $H$ if the characteristic polynomial of $G$ is divisible by that of $H$. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape…

For a positive integer $k$ and a graph $H$ on $k$ vertices, we are interested in the inducibility of $H$, denoted $\mathrm{ind}(H)$, which is defined as the maximum possible probability that choosing $k$ vertices uniformly at random from a…

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^-$, and an important step in their proof…

In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of…

The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning…

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

Let $T$ be a tree, a vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. The {\it reducible stem } of $T$ is the smallest subtree that contains all branch vertices of $T$. In…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…