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Pham Hoang Ha

Publications and source records attributed to Pham Hoang Ha.

At least 19 recordsLinked to original sources
arXiv2026

Degree sum conditions for a graph to have bounded conflict-free connection number

A path in an edge-coloured graph is called \emph{conflict-free} if a colour is exclusively applied to one of its edges. A graph $G$ is considered \emph{conflict-free connected} if every pair of vertices in $V(G)$ is connected by a conflict-free path. The minimum number of colours required to render a connected graph $G$ conflict-free connected is referred to as the \emph{conflict-free connection number}. In this paper, we introduce several sharp conditions on the minimum degree sum of any $4$ independent vertices in $G$ to ensure that the conflict-free connection number of $G$ is bounded.

math.CO↗
arXiv2026

The number of cut-edges and conflict-free connection number in planar graphs

A \textit{cut-edge} of a connected graph is an edge whose deletion increases the number of components. In this paper, we first state some conditions for a planar graph to have a few cut-edges. After that, we use the main results to study the conditions for a colored planar graph to have a bounded conflict-free connection number.

math.CO↗
arXiv2025

Spanning trees of claw-free graphs with few leaves and branch vertices

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 $T$. A graph is said to be claw-free if it does not contain $K_{1,3}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of claw-free graphs. Applying the main results, we also give some improvements of previous results on the spanning trees with few branch vertices for the case of claw-free graphs.

math.CO↗
arXiv2023

Spanning trees of $K_{1,4}$-free graphs whose reducible stems have few leaves

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 this paper, we give some sharp sufficient conditions for $K_{1,4}$-free graphs to have a spanning tree whose reducible stem having few leaves.

math.CO↗
arXiv2022

Spanning trees of $K_{1,4}$-free graphs with a bounded number of leaves and branch vertices

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 $T$. A graph is said to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_ {1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.

math.CO↗
arXiv2021

Spanning trees of a claw-free graph whose reducible stems have few leaves

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. For two distinct vertices $u,v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ For a leaf $x$ of $T$, let $y_x$ denote the nearest branch vertex to $x$. For every leaf $x$ of $T$, we remove the path $P_T [x, y_x)$ from $T$, where $P_T [x, y_x)$ denotes the path connecting $x$ to $y_x$ in $T$ but not containing $y_x$. The resulting subtree of $T$ is called the {\it reducible stem } of $T$. In this paper, we first use a new technique of Gould and Shull to state a new short proof for a result of Kano et al. on the spanning tree with a bounded number of leaves in a claw-free graph. After that, we use that proof to give a sharp sufficient condition for a claw-free graph having a spanning tree whose reducible stem has few leaves.

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arXiv2020

Progress on sufficient conditions for a graph to have a spanning $k-$ended tree

In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237] proved that if $G$ is a connected graph satisfying $σ_2(G) \geq |G|-k+1$ then $G$ has a spanning $k-$ended tree. They also gave an example to show that the condition "$σ_2(G) \geq |G|-k+1$" is sharp. In this paper, we introduce a new progress for this result. Let $K_{m,m+k}$ be a complete bipartite graph with bipartition $V(K_{m,m+k})=A\cup B, |A|=m, |B|=m+k.$ Denote by $H$ to be the graph obtained from $K_{m,m+k}$ by adding (or no adding) some edges with two end vertices in $A.$ We prove that if $G$ is a connected graph satisfying $σ_2(G) \geq |G|-k$ then $G$ has a spanning $k-$ended tree except for the case $G$ is isomorphic to a graph $H.$ As a corollary of our main result, a sufficient condition for a graph to have a few branch vertices is given.

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arXiv2018

Spanning trees with at most 2 branch vertices in claw - free graphs

In this article, we will prove that if $G$ is a connected claw-free graph and either $σ_6(G)\geq |G|-5$ or $σ_7(G)\geq |G|-2$, here $σ_k(G)$ is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a spanning tree with at most two branch vertices.

math.CO↗
arXiv2018

A note on independence number, connectivity and $k$-ended tree

A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $α_{G}(S) \leq k + κ_{G}(S)- 1.$ Then, $G$ has a $k$-ended tree which covers $S.$ Moreover, the condition is sharp.

math.CO↗
arXiv2018

Spanning trees with at most 4 leaves in $K_{1,5}-$free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with $σ_4(G)\geq n-1$ contains a spanning tree with at most $3$ leaves. In this paper, we prove an analogue of Kyaw's result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with $σ_5(G)\geq n-1$ contains a spanning tree with at most $4$ leaves. Moreover, the degree sum condition `$σ_5(G)\geq n-1$' is best possible.

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arXiv2018

A note on spanning trees of connected $K_{1,t}$-free graphs whose stems have a few leaves

Let $T$ be a tree, a vertex of degree one is called a leaf. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this note, we give a sharp sufficient condition to show that a $K_{1,t}-$free graph has a spanning tree whose stem has a few leaves. By applying the main result, we give improvements of previous related results.

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arXiv2018

Spanning trees in a Claw-free graph whose stems have at most $k$ branch vertices

Let $T$ be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this paper, we give two sufficient conditions for a connected claw-free graph to have a spanning tree whose stem has a bounded number of branch vertices, and those conditions are best possible. As corollaries of main results we also give some conditions to show that a connected claw-free graph has a spanning tree whose stem is a spider.

math.CO↗
arXiv2014

Ramification of the Gauss Map of Complete Minimal Surfaces in R^3 and R^4 on Annular Ends

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao.

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