Related papers: Unmixed and Cohen--Macaulay weighted oriented K\"o…

We recall a numerical criteria for Cohen--Macaulayness related to system of parameters, and introduce monomial ideals of K\"onig type which include the edge ideals of K\"onig graphs. We show that a monomial ideal is of K\"onig type if and…

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices.…

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is…

A graph G on omega_1 is called <omega-smooth if for each uncountable subset W of omega_1, G is isomorphic to G[W-W'] for some finite W'. We show that in various models of ZFC if a graph G is <omega-smooth then G is necessarily trivial, i.e,…

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper,…

There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform…

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar…

Given a 3-connected biased graph $\Omega$ with a balancing vertex, and with frame matroid $F(\Omega)$ nongraphic and 3-connected, we determine all biased graphs $\Omega'$ with $F(\Omega') = F(\Omega)$. As a consequence, we show that if $M$…

In this paper, we study rooted products of graphs from the perspective of combinatorial commutative algebra. For edge ideals, we introduce the 2-Cohen-Macaulayness with respect to a vertex and use it to investigate when edge ideals of…

We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of…

A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that…

We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$…

A hole is an induced cycle of length at least 4. Let $\l\ge 2$ be a positive integer, let ${\cal G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in {\cal G}_{\l}$. For…

We introduce and investigate the open neighborhood ideal $\mathcal{N}(G)$ of a finite simple graph $G$. We describe the minimal primary decomposition of $\mathcal{N}(G)$ in terms of the minimal total dominating sets (TDSs) of $G$. Then we…

Given a family of graphs $\mathcal{H}$, a graph $G$ is $\mathcal{H}$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to any graph in $\mathcal{H}$. We present sufficient and necessary conditions for a graph…

We denote a path on $t$ vertices as $P_t$ and a cycle on $t$ vertices as $C_t$. For two vertex-disjoint graphs $G_1$ and $G_2$, the {\em union} $G_1\cup G_2$ is the graph with $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$ and $E(G_1\cup…

We introduce a new family of graphs, namely, hybrid graphs. There are infinitely many hybrid graphs associated to a single graph. We show that every hybrid graph associated to a given graph is Cohen Macaulay. Furthermore, we show that every…

A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. A brick $G$ is solid if for any two vertex disjoint odd cycles $C_1$ and $C_2$ of $G$, $G-(V(C_1)\cup…

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph…

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an…