Spanning trees and spanning closed walks with small degrees

Let G be a graph and let f be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S?V(G)$, $\omega (G?S)<{\sum }_{v\in S}(f(v)?2)+2+\omega (G[S])$, then G has a spanning tree T containing an arbitrary given matching such that for each vertex v, $d_T(v)\le f(v)$, where $\omega (G?S)$ denotes the number of components of $G?S$ and $\omega (G[S])$ denotes the number of components of the induced subgraph $G[S]$ with the vertex set S. This is an improvement of several results. Next, we prove that if for all $S?V(G)$, $\omega (G?S)\le {\sum }_{v\in S}(f(v)?1)+1$, then G admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex v at most $f(v)$ times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).     

Path decompositions of triangle-free graphs

Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then $p(G)\le ?n/2?$. If G is allowed to be disconnected, then the upper bound $?\frac{3}{4}n?$ for $p(G)$ was obtained by Donald [7], which was improved to $?\frac{2}{3}n?$ independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, $?\frac{2}{3}n?$ is reached and so this bound is tight. If triangles are forbidden in G, then $p(G)\le ?\frac{g+1}{2g}n?$ can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that $p(G)\le ?3n/5?$, which improves the above result with $g=4$.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

Cyclotomic graphs and perfect codes

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[{\zeta }_m]/A$, with connection sets $\{\pm ({\zeta }_m^i+A):0\le i\le m?1\}$ and $\{\pm ({\zeta }_m^i+A):0\le i\le ?(m)?1\}$, respectively, where ${\zeta }_m$ ($m\ge 2$) is an mth primitive root of unity, A a nonzero ideal of $\mathbb{Z}[{\zeta }_m]$, and ? Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by $G_m(A)$ and $G_m^{?}(A)$, respectively. We give a necessary and sufficient condition for $D/A$ to be a perfect t-code in $G_m^{?}(A)$ and a necessary condition for $D/A$ to be such a code in $G_m(A)$, where $t\ge 1$ is an integer and D an ideal of $\mathbb{Z}[{\zeta }_m]$ containing A. In the case when $m=3,4$, $G_m((\alpha ))$ is known as an Eisenstein–Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for $(\beta )/(\alpha )$ to be a perfect t-code in $G_m((\alpha ))$, where $0\ne \alpha ,\beta \in \mathbb{Z}[{\zeta }_m]$ with β dividing α. In the literature such conditions are known to be sufficient when $m=4$ and $m=3$ under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.     

Irredundance trees of diameter 3

A set D of vertices of a graph $G=(V,E)$ is irredundant if each non-isolated vertex of $G[D]$ has a neighbour in $V?D$ that is not adjacent to any other vertex in D. The upper irredundance number $\mathrm{IR}(G)$ is the largest cardinality of an irredundant set of G; an $\mathrm{IR}(G)$-set is an irredundant set of cardinality $\mathrm{IR}(G)$.The IR-graph of G has the $\mathrm{IR}(G)$-sets as vertex set, and sets D and $D^{\prime }$ are adjacent if and only if $D^{\prime }$ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in $D^{\prime }$. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars $S(2n,2n)$, i.e., trees obtained by joining the central vertices of two disjoint stars $K_{1,2n}$.     

On Serre dimension of monoid algebras and Segre extensions

Let R be a commutative noetherian ring of dimension d and M be a commutative, cancellative, torsion-free monoid of rank r. Then S-$dim(R[M])\le max\{1,dim(R[M])?1\}$. Further, we define a class of monoids ${\{{\mathfrak{M}}_n\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal, then S-$dim(R[M])\le dim(R[M])?n=d+r?n$, where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over R, S-$dim(S_{mn}(R))\le dim(S_{mn}(R))?\left[\frac{m+n?1}{min\{m,n\}}\right]=d+m+n?1?\left[\frac{m+n?1}{min\{m,n\}}\right]$.     

On the Ramsey numbers of odd-linked double stars

The linked double star $S_c(n,m)$, where $n\ge m\ge 0$, is the graph consisting of the union of two stars $K_{1,n}$ and $K_{1,m}$ with a path on c vertices joining the centers. Its Ramsey number $r(S_c(n,m))$ is the smallest integer r such that every 2-coloring of the edges of a $K_r$ admits a monochromatic $S_c(n,m)$. In this paper, we study the Ramsey numbers of linked double stars when c is odd. In particular, we establish bounds on the value of $r(S_c(n,m))$ and determine the exact value of $r(S_c(n,m))$ if $n\ge c$, or if $n\le ?\frac{c}{2}??2$ and $m=2$.