arXiv subjects

Morteza Hasanvand

Publications and source records attributed to Morteza Hasanvand.

23 records · Page 2Linked to original sources

arXiv2021

The existence of uniform hypergraphs for which interpolation property of complete coloring fails

In 1967 Harary, Hedetniemi, and Prins showed that every graph $G$ admits a complete $t$-coloring for every $t$ with $χ(G) \le t \le ψ(G)$, where $χ(G)$ denotes the chromatic number of $G$ and $ψ(G)$ denotes the achromatic number of $G$ which is the maximum number $r$ for which $G$ admits a complete $r$-coloring. Recently, Edwards and Rz\c ażewski (2020) showed that this result fails for hypergraphs by proving that for every integer $k$ with $k\ge 9$, there exists a $k$-uniform hypergraph $H$ with a complete $χ(H)$-coloring and a complete $ψ(H)$-coloring, but no complete $t$-coloring for some $t$ with $χ(H)< t<ψ(H)$. They also asked whether there would exist such an example for $3$-uniform hypergraphs and posed another problem to strengthen their result. In this paper, we generalize their result to all cases $k$ with $k\ge 3$ and settle their problems by giving several kinds of $3$-uniform hypergraphs. In particular, we disprove a recent conjecture due to Matsumoto and the third author (2020) who suggested a special family of $3$-uniform hypergraph to satisfy the desired interpolation property.

math.CO↗

arXiv2019

Bipartite partition-connected factors with small degrees

In this paper, we show that every $2m$-partition-connected graph $G$ has a bipartite $m$-partition-connected factor $H$ such that for each vertex $v$, $d_H(v)\le \lceil \frac{3}{4}d_G(v)\rceil$. A graph $H$ is said to be $m$-partition-connected, if it contains $m$ edge-disjoint spanning trees. As an application, we conclude that tough enough graphs with appropriate number of vertices have a bipartite $m$-partition-connected factor with maximum degree at most $3m+1$. Finally, we prove that tough enough graphs of order at least $3k$ admit a bipartite connected factor whose degrees lie in the set $\{k,2k,3k,4k\}$.

math.CO↗

arXiv2018

Packing spanning rigid subgraphs with restricted degrees

Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. We say that $G$ is $l$-rigid, if it contains a spanning $l$-partition-connected subgraph $H$ with $|E(H)|=\sum_{v\in V(H)} l(v)-l(V(H))$. In this paper, we investigate decomposition of graphs into spanning partition-connected and spanning rigid subgraphs. As a consequence, we improve a recent result due to Gu (2017) by proving that every $(4kp-2p+2m)$-connected graph $G$ with $k\ge 2$ has a spanning subgraph $H$ containing a packing of $m$ spanning trees and $p$ spanning $(2k-1)$-edge-connected subgraphs $H_1,\ldots, H_p$ such that for each vertex $v$, every $H_i-v$ remains $(k-1)$-edge-connected and also $d_H(v)\le \lceil \frac{d_G(v)}{2}\rceil +2kp-p+m$. From this result, we refine a result on arc-connected orientations of graphs.

math.CO↗

arXiv2018

Packing spanning partition-connected subgraphs with small degrees

Let $G$ be a graph with $X\subseteq V(G)$ and let $l$ be an intersecting supermodular subadditive integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. Let $λ\in [0,1]$ be a real number and let $η$ be a real function on $X$. In this paper, we show that if $G$ is $l$-partition-connected and for all $S\subseteq X$, $$Θ_l(G \setminus S) \le \sum_{v\in S} (η(v) -2l(v))+l(V(G))+l(S)-λ(e_G(S))+l(S)),$$ then $G$ has an $l$-partition-connected spanning subgraph $H$ such that for each vertex $v\in X$, $d_H(v)\le \lceil η(v) -λl(v) \rceil $, where $e_G(S)$ denotes the number of edges of $G$ with both ends in $S$ and $Θ_l(G \setminus S)$ denotes the maximum number of all $\sum_{A\in P} l(A)-e_{G\setminus S}(P)$ taken over all partitions $P$ of $V(G)\setminus S$. Finally, we show that if $H$ is an $(l_1+\cdots +l_m)$-partition-connected graph, then it can be decomposed into $m$ edge-disjoint spanning subgraphs $H_1,\ldots, H_m$ such that every graph $H_i$ is $l_i$-partition-connected, where $l_1, l_2,\ldots, l_m$ are $m$ intersecting supermodular subadditive integer-valued functions on subsets of $V(H)$. These results generalize several known results.

math.CO↗

arXiv2017

Spanning closed walks with bounded maximum degrees of graphs on surfaces

Gao and Richter (1994) showed that every $3$-connected graph which embeds on the plane or the projective plane has a spanning closed walk meeting each vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995) extended this result to the torus and Klein bottle. Sanders and Zhao (2001) obtained a sharp result for higher surfaces by proving that every $3$-connected graph embeddable on a surface with Euler characteristic $χ\le -46$ admits a spanning closed walk meeting each vertex at most $\lceil \frac{6-2χ}{3}\rceil$ times. In this paper, we develop these results to the remaining surfaces with Euler characteristic $χ\le 0$.

math.CO↗