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July 1989 - present

## Publications

Publications (328)

Let L be list assignment of colors available for vertices of a graph G, an (L,r)-coloring of G is a proper coloring c such that for any vertex v∈V(G), we have c(v)∈L(v) and |c(N(v))|≥min{d(v),r}. The r-hued list chromatic number of G, denoted as χL,r(G), is the least integer k, such that for list assignment L satisfying |L(v)|=k ∀v∈V(G), G has an (...

Complete families of connected graphs, introduced by Catlin in the 1980s, have been known useful in the study of certain graphical properties that are closed under taking contractions. We show that given any complete family C of connected graphs such that C contains graphs with sufficiently many edge-disjoint spanning trees, for any real number a a...

Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ and a ${\mathbb Z}_p$-boundary $b$ of $G$, an orientation $\tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${\mathbb Z}_p$ at each vertex $v\in V(G)$ under orientation $D$. This concept was introduced by Esperet...

A (k,r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d,r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k,r)-coloring. This article is intended to survey the recent developments on the stud...

For a connected graph G, let κ′(G) be the edge-connectivity of G. The ℓ-edge-connectivity κℓ′(G) of G with order n≥ℓ is the minimum number of edges that are required to be deleted from G to produce a graph with at least ℓ components. It has been observed that while both κ′(G) and κℓ′(G) are related edge connectivity measures. In general, κℓ′(G) can...

For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank (1979) showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Baue...

Fault-tolerant networks are often modeled as s-hamiltonian graphs. Thus it is of interests to find graph families in which whether a graph is s-hamiltonian can be determined in polynomial time. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an hourglass-free graph is one that has no induced subgraph isomo...

The modulo orientation problem seeks a so-called mod (2t+1)-orientation of an undirected graph, in which the indegree is equal to outdegree under modulo 2t+1 at each vertex. Jaeger's circular flow conjecture states that every graph G with edge connectivity κ′(G)≥4t has a mod (2t+1)-orientation. Lovász et al. (2013) verified it for κ′(G)≥6t, and lat...

The index of a property P for a directed multigraph D is the smallest nonnegative integer k such that the iterated line digraph Lk(D) has the property P. Let e(D) denote the eulerian index of D and h(D) denote the hamiltonian index of D. Directed multigraphs families F and H are defined such that a directed multigraph D has a finite value e(D) if a...

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large n $n$, every 2‐edge‐connected simple graph G $G$ with n = ∣ V( G ) ∣ $n=| V(G)| $ and minimum degree δ( G ) ≥ n 5...

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if
the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a
connected simple graph $G$ that is not isomorphic to a path, a cycle, or a
$K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$
denote the smallest integer $i$ such that the...

Boesch and McHugh in [J. Combinatorial Theory Ser. B 38 (1985), 1-7] introduced the edge-maximal \((k, \ell )\)-graphs to study of network subcohesion, and obtained best possible upper size bounds for all edge-maximal \((k, \ell )\)-graphs. The best possible lower bounds are obtained in [J. Graph Theory 18 (1994), 227-240]. Let \(k,\ell > 0\) be in...

Let N1,1,1 be the graph formed by attaching a pendant edge to each vertex of a triangle, and B1,2 be a graph obtained by attaching end vertices of two disjoint paths of lengths 1,2 to two vertices of a triangle. Broersma (1993) [2] and Čada et al. (2016) [3] conjectured that for a 2-connected claw-free simple graph G and for a fixed graph Γ∈{N1,1,1...

For a hamiltonian property P, Clark and Wormold introduced the problem of investigating the value P(a,b)=max{min{n:Ln(G) has property P}: κ′(G)≥a and δ(G)≥b}, and proposed a few problems to determine P(a,b) with b≥a≥4 when P is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-conne...

Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number α(G)≤k−s, then G is Hamilton-connected (s=1) or Hamiltonian (s=0) or traceable (s=−1). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamilt...

For a given list assignment L of a graph G, an (L,r)-coloring of G is a proper coloring c such that for any vertex v with degree d(v), v is adjacent to vertices of at least min{d(v),r} different color with c(v)∈L(v). The r-hued list chromatic number of G, denoted as χL,r(G), is the least integer k, such that for any v∈V(G) and every list assignment...

For a digraph [Formula: see text], if [Formula: see text] contains a spanning closed trail, then [Formula: see text] is supereulerian. If for any pairs vertices [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text] contains both a spanning [Formula: see text]-trail and a spanning [Formula: see text]-trail, then [Fo...

A graph is supereulerian if it has a spanning eulerian subgraph. Harary and Nash-Williams in 1968 proved that the line graph of a graph G is hamiltonian if and only if G has a dominating eulerian subgraph, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint span...

A graph G is Hamilton-connected if for any pair of distinct vertices \(u, v \in V(G)\), G has a spanning (u, v)-path; G is 1-hamiltonian if for any vertex subset \(S \subseteq {V(G)}\) with \(|S| \le 1\), \(G - S\) has a spanning cycle. Let \(\delta (G)\), \(\alpha '(G)\) and L(G) denote the minimum degree, the matching number and the line graph of...

For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in 1979 showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. We investigate the value of the...

Let D be a digraph and let α(D), α′(D) and λ(D) be independence number, the matching number and the arc-strong connectivity of D, respectively. Bang-Jensen and Thommassé in 2011 conjectured that every digraph D with λ(D)≥α(D) is supereulerian. In [J. Graph Theory, 81(4), (2016) 393-402], it is shown that every digraph D with λ(D)≥α′(D) is supereule...

A digraph D is supereulerian if D has a spanning closed trail, and is strongly trail-connected if for any pair of vertices u,v∈V(D), D has a spanning (u,v)-trail and a spanning (v,u)-trail. The symmetric core J=J(D) of a digraph D is a spanning subdigraph of D with A(J) consisting of all symmetric arcs in D. Let J1,J2,⋯,Jk(D) be the connected symme...

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the...

The spanning tree packing number of a graph G is the maximum number of edge-disjoint spanning trees in G, and the arboricity of G is the minimum number of edge-disjoint forests needed to partition the edge set of G. In this paper, we give bounds on the spanning tree packing number and the arboricity of graphs in terms of effective resistances. As a...

Let r2≥r1≥0 be two integers. A bipartite graph G is two-disjoint-cycle-cover vertex [r1,r2]-bipancyclic (2-DCC vertex [r1,r2]-bipancyclic in short) if for any two vertices u,v∈V(G) and any even integer ℓ satisfying r1≤ℓ≤r2, there exist two vertex-disjoint cycles J1 and J2 in G with |V(J1)|=ℓ and |V(J2)|=|V(G)|−ℓ such that u∈V(J1) and v∈V(J2); and t...

A graph is supereulerian if it contains a spanning closed trail. We prove several Erds-type extremal size conditions with a lower bounded minimum degree for a graph to be supereulerian and with different edge-connectivity, with the corresponding extremal graphs characterized. These results are then applied to prove sufficient conditions involving a...

Recently, Huang showed that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional hypercube has maximum degree at least $\sqrt{n}$ in [Annals of Mathematics, 190 (2019), 949--955]. In this paper, we discuss the induced subgraphs of Cartesian product graphs and semi-strong product graphs to generalize Huang's result. Let $\Gamma_1$ be a...

For a graph G, an integer s≥0 and distinct vertices u,v∈V(G), an (s;u,v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. The spanning connectivity κ∗(G) is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with u≠v, G has a spanning (k;u,v)-path-system. It is known that κ∗(G)≤κ(G), and deter...

For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete....

A fractional matching of a graph G is a function f : E ( G ) → [ 0 , 1 ] such that for any v ∈ V ( G ) , ∑ e ∈ E G ( v ) f ( e ) ≤ 1 where E G ( v ) = { e ∈ E ( G ) : e is incident with v in G } . The fractional matching number of G is μ f ( G ) = max { ∑ e ∈ E ( G ) f ( e ) : f is a fractional matching of G } . For any real numbers a ≥ 0 and k ∈ (...

A graph is supereulerian if it has a spanning eulerian subgraph. We show that a connected simple graph G with n=|V(G)|≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$...

A graph G is strongly spanning trailable if for any \(e_1=u_1v_1, e_2=u_2v_2\in E(G)\) (possibly \(e_1=e_2\)), \(G(e_1, e_2)\), which is obtained from G by replacing \(e_1\) by a path \(u_1v_{e_1}v_1\) and by replacing \(e_2\) by a path \(u_2v_{e_2}v_2\), has a spanning \((v_{e_1}, v_{e_2})\)-trail. A graph G is Hamilton-connected if there is a spa...

For integers s1,s2,s3>0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results.
(i) Let N1={Ns1,s2,s3:s1>0,s1≥s2≥s3≥0 and s1+s2+s3≤6}. Then for any N∈N1, every N-free line graph L(G) with |V(L(G))|...

A graph G is pancyclic if it contains cycles of all possible lengths. A graph G is 1-hamiltonian if the removal of at most 1 vertices from G results in a hamiltonian graph. In Veldman (1988) Veldman showed that the line graph L(G) of a connected graph G with diameter at most 2 is hamiltonian. In this paper, we continue studying the line graph L(G)...

A (k, r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d, r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k, r)-coloring. Let f(r)=r+3 if 1 ≤ r ≤ 2, f(r)=r+5 if 3 ≤ r ≤ 7 and f(r)=⌊3r/2⌋+1...

For a graph G, the flow index ϕ(G) is the smallest rational number t>0 such that the graph has a circular t-flow. Li et al. (2018) recently proved that ϕ(G)<3 for any 8-edge-connected graph G, and conjectured that 6-edge-connectivity would suffice. Here we present a contraction method to investigate this problem and apply it to verify this conjectu...

Esperet, de Joannis de Verclos, Le and Thomassé in [SIAM J. Discrete Math., 32(1) (2018), 534–542] introduced the problem that for an odd prime p, whether there exists an orientation D of a graph G for any mapping f:E(G)→Zp∗ and any Zp-boundary b of G, such that under D, at every vertex, the net out f-flow is the same as b(v) in Zp. Such an orienta...

Let G be a graph and let κ(G) be the vertex-connectivity of G. The maximum subgraph connectivity of G is κ¯(G)=max{κ(H):H⊆G}. A simple graph G is vertex-k-maximal if κ¯(G)≤k, but for any e∈E(Gc), κ¯(G+e)≥k+1. Mader conjectured that every vertex-k-maximal simple graph of order n satisfies |E(G)|≤32(k−13)(n−k).
We prove the following. (i) Every verte...

Let c(G), g(G), ω(G) and μn−1(G) denote the number of components, the girth, the clique number and the second smallest Laplacian eigenvalue of the graph G, respectively. The strength η(G) and the fractional arboricity γ(G) are defined byη(G)=minF⊆E(G)|F|c(G−F)−c(G)andγ(G)=maxH⊆G|E(H)||V(H)|−1, where the optima are taken over all edge subsets F an...

Let κ′(G), μn−1(G) and μ1(G) denote the edge-connectivity, the algebraic connectivity and the Laplacian spectral radius of G, respectively. In this paper, we prove that for integers k≥2 and r≥2, and any simple graph G of order n with minimum degree δ≥k, girth g≥3 and clique number ω(G)≤r, the edge-connectivity κ′(G)≥k if μn−1(G)≥(k−1)nN(δ,g)(n−N(δ,...

A graph is supereulerian if it has a spanning closed trail. Catlin in 1990 raised the problem of determining the reduced nonsupereulerian graphs with small orders, as such results are of particular importance in the study of Eulerian subgraphs and Hamiltonian line graphs. We determine all reduced graphs with order at most 14 and with few vertices o...

Let k,n,s,t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k,n, s, t > 0$$\end{document} be integers and n=s+t≥2k+2\documentclass[12pt]{minimal} \usepackage{amsmath}...

Let \(\alpha '(G), ess'(G), \kappa (G), \kappa '(G), N_G(v)\) and \(D_i(G)\) denote the matching number, essential edge connectivity, connectivity, edge connectivity, the set of neighbors of v in G and the set of degree i vertices of a graph G, respectively. For \(u, v\in V(G)\), define \(u\sim v\) if and only if \(u=v\) or both \(u, v\in D_{2}(G)\...

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is $\mu_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$ is fractional matching of $G\}$. For any real numbers $a \ge 0$ and...

Let $\kappa'(G)$, $\kappa(G)$, $\mu_{n-1}(G)$ and $\mu_1(G)$ denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of $G$, respectively. In this paper, we prove that for integers $k\geq 2$ and $r\geq 2$, and any simple graph $G$ of order $n$ with minimum degree $\delta\geq k$, girth $g\geq 3...

Let H = (V, E) be a hypergraph, where V is a set of vertices and E is a set of non-empty subsets of V called edges. If all edges of H have the same cardinality r, then H is an r-uniform hypergraph; if E consists of all r-subsets of V, then H is a complete r-uniform hypergraph, denoted by Krn, where n = |V |. An r-uniform hypergraph H = (V, E) is (k...

The line graph L(G) of a graph G is a simple graph with E(G) being its vertex set, where two vertices are adjacent in L(G) whenever the corresponding edges share a common vertex in G. A graph H is even if every vertex of H has even degree, and a graph is supereulerian if it has a spanning closed trail. We obtain a characterization for a graph G to...

A matroid M with a distinguished element e0∈E(M) is a rooted matroid with e0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely m...

A digraph D is supereulerian if D has a spanning eulerian subdigraph. We investigate forbidden induced subdigraph con- ditions for a strong digraph to be supereulerian. Let Pk denote the dipath on k vertices. For k ∈ {2, 3, 4}, we determine the smallest integer hk such that if a strong strict digraph D containing a subdigraph H isomorphic to Pk alw...

Let H=(V,E) be a hypergraph, where V is a set of vertices and E is a set of non-empty subsets of V called edges. If all edges of H have the same cardinality r, then H is a r-uniform hypergraph; if E consists of all r-subsets of V, then H is a complete r-uniform hypergraph, denoted by Knr, where n=|V|. A hypergraph H′=(V′,E′) is called a subhypergra...

Let τ(G) and κ′(G) denote the spanning tree packing number and the edge-connectivity of a graph G, respectively. Cioabă and Wong (2012) in [5] conjectured an explicit relationship between τ(G) and the second largest adjacency eigenvalue λ2(G) of a regular graph. Gu et al. (2016) in [12] presented a more general conjecture on a simple graph G. This...

Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be $k$-connected and maximally connected, respectively. Similar upper bounds of $q(G)$ and lower bounds of $q(\overl...

In connection to the 5-flow conjecture, a modulo 5-orientation of a graph G is an orientation of G such that the indegree is congruent to outdegree modulo 5 at each vertex. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte's 5-flow conjecture. In this paper, we study the problem o...

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger’s result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connecte...

Let τ(G) be the maximum number of mutually edge-disjoint spanning trees contained in a graph G and let κ′(G) denote the edge-connectivity of G. As a corollary of the spanning trees packing theorem by Nash-Williams and Tutte, it is known that if κ′(G)≥2k, then τ(G)≥k. An edge-cut X of G is an essential edge-cut if G−X contains at least two nontrivia...

A graph G is said to be pancyclic if G contains cycles of lengths from 3 to |V(G)|. For a positive integer i, we use \(Z_i\) to denote the graph obtained by identifying an endpoint of the path \(P_{i+1}\) with a vertex of a triangle. In this paper, we show that every 4-connected claw-free \(Z_8\)-free graph is either pancyclic or is the line graph...

Two graphs are adjacency cospectral (respectively, permanental cospectral) if they have the same adjacency spectrum (respectively, permanental spectrum). In this paper, we present a new method to construct new adjacency cospectral and permanental cospetral pairs of graphs from smaller ones. As an application, we obtain an infinite family of pairs o...

A proper k-coloring c of a graph G is a -coloring if for every vertex v with degree there are at least min different colors present in the neighborhood of v. The r-hued chromatic number of G, , is the least integer k such that G has a -coloring. We show that, for any , there exist infinitely many graphs G with the property that G contains a subgrap...

A graph $G$ is said to be Hamiltonian (resp., traceable) if it has a cycle (resp., path) that contains every vertex of $G$. A bipartite graph $G=[U,V]$ is balanced if $|U|=|V|$. There have been researches on the relationship between graph eigenvalues and the hamiltonian properties and path-coverable properties of the graph. We prove spectral theore...

Let $\lambda_{1}(G)$ and $\mu_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $\lambda_{1}(G)$ based on the connectivity to assure a connected graph to be Hamiltonian and traceable, respectively. In this paper, we p...

Let $\tau(G)$ and $\kappa'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $\delta \ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $\lambda_2(G)...

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A hypergraph $H...

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A $r$-uniform h...

A mod (2p+1)-orientation D is an orientation of G such that dD⁺(v)≡dD⁻(v)(mod2p+1) for any vertex v∈V(G). Extending Tutte's integer flow conjectures, it was conjectured by Jaeger that every 4p-edge-connected graph has a mod (2p+1)-orientation. However, this conjecture has been disproved in Han et al. (2018) recently. Infinite families of 4p-edge-co...

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of elements called vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$....

A digraph D is supereulerian if D has a spanning closed ditrail. Bang-Jensen and Thomassé conjectured that if the arc-strong connectivity λ(D) of a digraph D is not less than the independence number α(D), then D is supereulerian. A digraph is bipartite if its underlying graph is bipartite. Let α′(D) be the size of a maximum matching of D. We prove...

Let λ(D) be the arc strong-connectivity of a digraph D, and k>0 be an integer, and α,β be rational numbers. A strong digraph D is locally (α,β)+-arc-connected if v V(D), λ(D[N+(v)])≥α|N+(v)|+β. A locally (0,k)+-arc-connected digraph is also called k+-locally-arc-connected. We show that for any integer k, a strong, k+-locally-arc-connected digraph m...

A mod (2p + 1)-orientation D is an orientation of G such that d⁺D(v) − d⁻D(v) ≡ 0 (mod 2p + 1) for any vertex v ∈ V (G). Jaeger conjectured that every 4p-edge-connected graph has a mod (2p + 1)-orientation. A graph G is strongly Z2p+1-connected if for every mapping b : V (G) → Z2p+1 withv∈V (G) b(v) = 0, there exists an orientation D of G such that...

Given a zero-sum function β:V(G)→Z3 with ∑v∈V(G)β(v)=0, an orientation D of G with dD+(v)−dD−(v)=β(v) in Z3 for every vertex v∈V(G) is called a β-orientation. A graph G is Z3-connected if G admits a β-orientation for every zero-sum function β. Jaeger et al. conjectured that every 5-edge-connected graph is Z3-connected. A graph is ⟨Z3⟩-extendable at...

For a digraph D, let $\lambda (D)$ be the arc-strong-connectivity of D. For an integer $k > 0$, a simple digraph D with $|V(D)| \ge k+1$ is k-maximal if every subdigraph H of D satisfies $\lambda (H) \le k$ but for adding new arc to D results in a subdigraph $H'$ with $\lambda (H') \ge k + 1$. We prove that if D is a simple k-maximal digraph on $n...

Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. The conjecture has been verified for paths, trees...

For a graph G, the supereulerian width μ'(G) of a graph G is the largest integer s such that G has a spanning (k;u,v)-trail-system, for any integer k with 1≤k≤s, and for any u,v∈V(G) with u≠v. Thus μ'(G)≥2 implies that G is supereulerian, and so graphs with higher supereulerian width are natural generalizations of supereulerian graphs. Settling an...

In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ i...

For positive integers k and r, a (k,r)-coloring of a graph G is a proper coloring of the vertices with k colors such that every vertex of degree i will be adjacent to vertices with at least min{i,r} different colors. The r-dynamic chromatic number of G, denoted by χr(G), is the smallest integer k for which G has a (k,r)-coloring. For a k-list assig...

Let ℤm be the cyclic group of order m ≥ 3. A graph G is ℤm-connected if G has an orientation D such that for any mapping b : V (G) → ℤm with ΣvϵV(G)b(v) = 0, there exists a mapping f : E(G) → ℤm - 0 satisfying ΣeϵE-D(v)f(e) - ΣeϵE- D(v)f(e) = b(v) in ℤm for any v ϵ V (G); and a graph G is strongly ℤm-connected if, for any mapping θ : V (G) → ℤm wit...

Given a zero-sum function $\beta : V(G) \rightarrow \mathbb{Z}_3$ with $\sum_{v\in V(G)}\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \beta(v)$ in $\mathbb{Z}_3$ for every vertex $v\in V(G)$ is called a $\beta$-orientation. A graph $G$ is $\mathbb{Z}_3$-connected if $G$ admits a $\beta$- orientation for every zero-sum function $\b...

For a digraph D, let lambda(D) be the arc-strong-connectivity of D. For an integer k > 0, a simple digraph D with vertical bar V (D)vertical bar >= k + 1 is k-maximal if every subdigraph H of D satisfies lambda(H) <= k but for adding new arc to D results in a subdigraph H' with lambda(H') >= k + 1. We prove that if D is a simple k-maximal digraph o...

In a partial inverse matroid problem, given a matroid , a real valued weight function _w_ on _S_, and an independent set , the goal is to modify the weight _w_ as small as possible to a new weight such that there exists a -maximum base containing . In this paper, we study a constraint version of the partial inverse matroid problem in which the weig...

In a partial inverse combinatorial problem, given a partial solution, the goal is to modify data as small as possible such that there exists an optimal solution containing the given partial solution. In this paper, we study a constraint version of the partial inverse matroid problem in which the weight can only be increased. Two polynomial time alg...

A digraph D is supereulerian if D has a spanning directed eulerian subdigraph. Hong et al. proved that delta(+)(D) + delta(-)(D) >= vertical bar V(D)vertical bar - 4 implies D is supereulerian except some well-characterized digraph classes if the minimum degree is large enough. In this paper, we characterize the digraphs D which are not supereuleri...

Let denote the class of c-cyclic graphs with n vertices, girth and pendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connected c-cyclic graphs with vertices, girth g and at most pendant vertices, respectively, and the unique extremal...

Settling a conjecture of Kuipers and Veldman posted in Favaron and Fraisse (2001) [9], Lai et al. (2006) [15] proved that if H is a 3-connected claw-free simple graph of order n≥196, and if δ(H)≥n+510, then either H is Hamiltonian, or the Ryjáček's closure cl(H)=L(G) where G is the graph obtained from the Petersen graph P by adding n−1510 pendant e...

Catlin in 1988 indicated that there exist graph families such that if every edge e in a connected graph G lies in a subgraph of G isomorphic to a member in , then G is supereulerian. In particular, if every edge of a connected graph G lies in a 3-cycle, then G is supereulerian. The purpose of this research is to investigate how Catlin's theorem can...

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow (Formula presented.) of the contracted graph...

A proper vertex -coloring of a graph is dynamic if for every vertex with degree at least 2, the neighbors of receive at least two different colors. The smallest integer such that has a dynamic -coloring is the dynamic chromatic number . In this paper the differences between and , and between and are investigated respectively.

We investigate the relationship between the eigenvalues of a graph G and fractional spanning tree packing and coverings of G. Let ω(G) denote the number of components of a graph G. The strength η(G) and the fractional arboricity γ(G) are defined by η(G)=min|X|ω(G-X)-ω(G),andγ(G)=max|E(H)||V(H)|-1, where the optima are taken over all edge subsets X...

Let G be a graph and s>0 be an integer. If, for any function b:V(G)→Z2s+1 satisfying ∑v∈V(G)b(v)≡0(mod2s+1), G always has an orientation D such that the net outdegree at every vertex v is congruent to b(v) mod 2s+1, then G is strongly Z2s+1-connected. For a graph G, denote by α(G) the cardinality of a maximum independent set of G. In this paper, we...

Let be a sequence of of nonnegative integers pairs. If a digraph D with satisfies and for each i with , then d is called a degree sequence of D. If D is a strict digraph, then d is called a strict digraphic sequence. Let be the collection of digraphs with degree sequence d. We characterize strict digraphic sequences d for which there exists a stric...