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Baoyindureng Wu

Publications and source records attributed to Baoyindureng Wu.

At least 19 recordsLinked to original sources
arXiv2026

Counterexamples to two conjectures on modular edge colorings of graphs

For an integer $k\geq2$, let $χ_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897--904), which states that $χ_k'(G)\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\{k,2k\}$ and a specified vertex partition. With a suitable choice of the part sizes, if the number of edges inside one part is $o(k^2)$, then $χ_k'(G)\geq(4-2\sqrt2+o(1))k$. This gives connected bipartite and connected nonbipartite counterexamples. In particular, the same examples also disprove the earlier conjecture of Botler, Colucci, and Kohayakawa (The mod $k$ chromatic index of graphs is $O(k)$, J. Graph Theory 102 (2023) 197--200), which states that $χ_k'(G)\leq k+C$ for some absolute constant $C$.

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arXiv2025

Odd spanning trees of a graph

A graph $G=(V,E)$ is said to be odd (or even, resp.) if $d_G(v)$ is odd (or even, resp.) for any $v\in V$. Trivially, the order of an odd graph must be even. In this paper, we show that every 4-edge connected graph of even order has a connected odd factor. A spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST by simply) if $T$ contains no vertex of degree two. Trivially, an odd spanning tree must be a HIST. In 1990, Albertson, Berman, Hutchinson, and Thomassen showed that every connected graph of order $n$ with $δ(G)\geq \min\{\frac n 2, 4\sqrt{2n}\}$ contains a HIST. We show that every complete bipartite graph with both parts being even has no odd spanning tree, thereby for any even integer $n$ divisible by 4, there exists a graph of order $n$ with the minimum degree $\frac n 2$ having no odd spanning tree. Furthermore, we show that every graph of order $n$ with $δ(G)\geq \frac n 2 +1$ has an odd spanning tree. We also characterize all split graphs having an odd spanning tree. As an application, for any graph $G$ with diameter at least 4, $\overline{G}$ has a spanning odd double star. Finally, we also give a necessary and sufficient condition for a triangle-free graph $G$ whose complement contains an odd spanning tree. A number of related open problems are proposed.

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arXiv2024

Maximum odd induced subgraph of a graph concerning its chromatic number

Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2χ(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $χ(G)$ is the chromatic number of $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang and Wargo in 1997, which states that $f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor$ for a connected graph $G$ of order $n$. Scott's conjecture is open for a graph with chromatic number at least 3.

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arXiv2023

Gallai's path decomposition conjecture for cartesian product of graphs (\uppercase\expandafter{\romannumeral 2})

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. In this paper, we prove that Gallai's path decomposition conjecture holds for the cartesian product $G\Box H$, where $H$ is any graph and $G$ is a unicyclic graph or a bicyclic graph.

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arXiv2023

Gallai's path decomposition conjecture for Cartesian product of graphs

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. Let $n_o(G)$ to denote the number of vertices with odd degree in $G$. Lovász proved that if $G$ is a connected graph with all vertices having degree odd, i.e. $n_o(G)=n$, then $p(G)=\frac n 2$. In this paper, we prove that if $G$ is a connected graph of order $m\geq 2$ with $p(G)=\frac{n_o(G)}{2}$ and $H$ is a connected graph of order $n$, then $p(G\Box H)\leq\frac{mn}{2}$. Furthermore, we prove that $p(G)=\frac{n_o(G)}{2}$, if one of the following is hold: (\romannumeral1) $G$ is a tree; (\romannumeral2) $G=P_n\Box T$, where $n\geq 4$ and $T$ is a tree; (\romannumeral3) $G=P_n\Box H$, where $H$ is an even graph.

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arXiv2020

On the arithmetic-geometric index of graphs

Very recently, the first geometric-arithmetic index $GA$ and arithmetic-geometric index $AG$ were introduced in mathematical chemistry. In the present paper, we first obtain some lower and upper bounds on $AG$ and characterize the extremal graphs. We also establish various relations between $AG$ and other topological indices, such as the first geometric-arithmetic index $GA$, atom-bond-connectivity index $ABC$, symmetric division deg index $SDD$, chromatic number $χ$ and so on. Finally, we present some sufficient conditions of $GA(G)>GA(G-e)$ or $AG(G)>AG(G-e)$ for an edge $e$ of a graph $G$. In particular, for the first geometric-arithmetic index, we also give a refinement of Bollobás-Erdős-type theorem obtained in [3].

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arXiv2020

Eigenvalues and triangles in graphs

Bollobás and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $λ^2_1(G)+λ^2_2(G)\leq \frac{r-1}{r}\cdot2m$, where $λ_1(G)$ and $λ_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $λ_1(G)\geq\sqrt{m-1}$ and $G\neq C_5\cup (n-5)K_1$; and (2) $λ_1(G)\geq λ_1(S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}))$ and $G\neq S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$, where $S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$ is obtained from $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

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arXiv2017

Feedback vertex number of Sierpiński-type graphs

The feedback vertex number $τ(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $τ(S_p^n)=p^{n-1}(p-2)$ for the Sierpiński graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpiński triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpiński graph $S_p^{n+1}$. We prove that $τ(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{|V(\hat{S}_3^n)|} 3$, and give an upper bound for $τ(\hat{S}_p^n)$ for the case when $p\geq 4$.

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arXiv2017

Connected even factors in the square of essentially 2-edge connected graphs

In this paper we prove that the square of an essentially 2-edge connected graph with an additional property has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge connected graph does not contain a connected even factor with bounded maximum degree.

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arXiv2017

On the maximum value of conflict-free verex-connection number of graphs

A path in a vertex-colored graph is called {\it conflict-free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be {\it conflict-free vertex-connected} if any two vertices of the graph are connected by a conflict-free path. The {\it conflict-free vertex-connection number}, denoted by $vcfc(G)$, is defined as the smallest number of colors required to make $G$ conflict-free vertex-connected. Li et al. conjectured that for a connected graph $G$ of order $n$, $vcfc(G)\leq vcfc(P_n)$. We confirm that the conjecture is true and pose a a relevant conjecture concerning the conflict-free connection number introduced by Czap et al..

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arXiv2017

A note on edge degree and spanning trail containing given edges

Let $G$ be a simple graph with $n\geq4$ vertices and $d(x)+d(y)\geq n+k$ for each edge $xy\in E(G)$. In this work we prove that $G$ either contains a spanning closed trail containing any given edge set $X$ if $|X|\leq k$, or $G$ is a well characterized graph. As a corollary, we show that line graphs of such graphs are $k$-hamiltonian.

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arXiv2017

Independent transversal domination number of a graph

Let $G=(V, E)$ be a graph. A set $S\subseteq V(G)$ is a {\it dominating set} of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex of $S$. The {\it domination number} of $G$, denoted by $γ(G)$, is the cardinality of a minimum dominating set of $G$. Furthermore, a dominating set $S$ is an {\it independent transversal dominating set} of $G$ if it intersects every maximum independent set of $G$. The {\it independent transversal domination number} of $G$, denoted by $γ_{it}(G)$, is the cardinality of a minimum independent transversal dominating set of $G$. In 2012, Hamid initiated the study of the independent transversal domination of graphs, and posed the following two conjectures: Conjecture 1. If $G$ is a non-complete connected graph on $n$ vertices, then $γ_{it}(G)\leq\lceil\frac{n}{2}\rceil$. Conjecture 2. If G is a connected bipartite graph, then $γ_{it}(G)$ is either $γ(G)$ or $γ(G)+1$. We show that Conjecture 1 is not true in general. Very recently, Conjecture 2 is partially verified to be true by Ahangar, Samodivkin, Yero. Here, we prove the full statement of Conjecture 2. In addition, we give a correct version of a theorem of Hamid. Finally, we answer a problem posed by Martínez, Almira, and Yero on the independent transversal total domination of a graph.

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arXiv2016

A $\overrightarrow{P_{3}}$-decomposition of tournaments and bipartite digraphs

A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a $\overrightarrow{P_{3}}$-decomposition. This solves a problem posed by Diwan ($\overrightarrow{P_{3}}$-decomposition of directed graphs, Discrete Appl. Math., http:// dx.doi.org/10.1016/j.dam.2016.01.039.).

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arXiv2016

Two-log-convexity of the Catalan-Larcombe-French sequence

The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence $\{P^2_n-P_{n-1}P_{n+1}\}_{n\geq 1}$ are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.

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arXiv2016

Construction and characterization of graphs whose each spanning tree has a perfect matching

An edge subset $S$ of a connected graph $G$ is called an anti-Kekulé set if $G-S$ is connected and has no perfect matching. We can see that a connected graph $G$ has no anti-Kekulé set if and only if each spanning tree of $G$ has a perfect matching. In this paper, by applying Tutte's 1-factor theorem and structure of minimally 2-connected graphs, we characterize all graphs whose each spanning tree has a perfect matching In addition, we show that if $G$ is a connected graph of order $2n$ for a positive integer $n\geq 4$ and size $m$ whose each spanning tree has a perfect matching, then $m\leq \frac{(n+1)n} 2$, with equality if and only if $G\cong K_n\circ K_1$.

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arXiv2015

More bounds for the Grundy number of graphs

A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every $j<i$. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number $Γ(G)$ of a graph $G$ is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new characterization for a $\{P_{4}, C_4\}$-free graph by supporting a conjecture of Zaker, which says that $Γ(G)\geq δ(G)+1$ for any $C_4$-free graph $G$.

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arXiv2015

Upper bounds for the achromatic and coloring numbers of a graph

Dvořák \emph{et al.} introduced a variant of the Randić index of a graph $G$, denoted by $R'(G)$, where $R'(G)=\sum_{uv\in E(G)}\frac 1 {\max\{d(u), d(v)\}}$, and $d(u)$ denotes the degree of a vertex $u$ in $G$. The coloring number $col(G)$ of a graph $G$ is the smallest number $k$ for which there exists a linear ordering of the vertices of $G$ such that each vertex is preceded by fewer than $k$ of its neighbors. It is well-known that $χ(G)\leq col(G)$ for any graph $G$, where $χ(G)$ denotes the chromatic number of $G$. In this note, we show that for any graph $G$ without isolated vertices, $col(G)\leq 2R'(G)$, with equality if and only if $G$ is obtained from identifying the center of a star with a vertex of a complete graph. This extends some known results. In addition, we present some new spectral bounds for the coloring and achromatic numbers of a graph.

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arXiv2015

Remoteness and distance eigenvalues of a graph

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $ρ$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $ρ+\partial_3>0$ when $d\geq 3$ and $ρ+\partial_{\lfloor\frac{7d}{8}\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\partial_n+ρ$ and $\partial_1-ρ$ when $G\ncong K_n$ and the extremal graphs are characterized.

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