arXiv subjects

Fuliang Lu

Publications and source records attributed to Fuliang Lu.

At least 19 recordsLinked to original sources

arXiv2026

Bricks that every removable edge is solitary

A brick is a 3-connected graph $G$ such that $G-u-v$ has a perfect matching for any two distinct vertices $u,v\in V(G)$. An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is matching covered. We say that a removable edge $e$ in a brick $G$ is $b$-invariant if $b(G-e)=b(G)=1$, where $b(H)$ denotes the number of bricks in the tight cut decomposition of a matching covered graph $H$. An edge of a graph is solitary if it lies in precisely one perfect matching. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. Note that every $b$-invariant edge is removable. In this paper, we strengthen the condition by requiring that every removable edge is solitary. We show that every nonsolid brick satisfying this strengthened condition can be obtained by repeatedly splicing odd wheels (up to multiple edges). Moreover, properties of such bricks imply that "repeatedly splicing odd wheels" cannot be replaced by "repeatedly splicing copies of $K_4$".

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arXiv2026

The number of perfect matchings in 3-connected planar graphs

A graph is matchable if it admits a perfect matching. Recently, Goedgebeur et al. asked whether there exists a constant $c<12$ such that infinitely many matchable planar $3$-connected graphs, each with exactly $c$ perfect matchings. We answer this question by proving that every matchable planar $3$-connected graph on at least 40 vertices has at least 12 perfect matchings, and this lower bound is sharp.

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arXiv2026

Near-bipartite bricks in which every b-invariant edge is a forcing edge

A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lov\'asz proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges. Denote by b(G) the number of bricks in such a decomposition. An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G). Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from K_4, \overline{C_6}, and the Petersen graph, in which every b-invariant edge is a forcing edge. In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.

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arXiv2026

Bricks in which every vertex is incident with a forcing edge

A matching covered graph is a brick if it is 3-connected and bicritical. An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. We prove that every vertex of a simple brick G is incident with a forcing edge if and only if G is an odd wheel or is isomorphic to one of the following three graphs: the triangular prism \overline{C_6},\overline{C_6}^+ or the bicorn H_8. Here \overline{C_6}^+ is obtained from \overline{C_6} by adding an edge between two nonadjacent vertices.

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arXiv2026

Tight cuts in matching covered graphs

An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and Pulleyblank [Brick decompositions and the matching rank of graphs, Combinatorica 2(3) (1982) 247-274] proved that if a matching covered graph has a non-trivial tight cut, then it also has a non-trivial ELP-cut. In confirmation of a conjecture proposed by Carvalho, Lucchesi and Murty, Chen et. al. [Laminar tight cuts in matching covered graphs, J. Comb. Theory, Ser. B, 150 (2021) 177-194] showed that if C is a non-trivial tight cut of a matching covered graph G, then G has at least one C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C. In this paper, we present a complete characterization of non-trivial tight cuts in matching covered graphs, from which the result of Chen et. al. can be derived directly. Moreover, we show that the lower bound of the number of C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C in the result of Chen et. al. is sharp.

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arXiv2026

Adjacent vertices of small degree in minimal matching covered graphs

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An edge is called a $k$-line if both of its end vertices are of degree $k$. Lov\'asz and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] proved that a minimal matching covered bipartite graph different from $K_2$ has minimum degree 2 and contains at least $[(|V(G)|+15)/6]$ 2-lines by ear decompositions. He et al. [J. Graph Theory 111 (2026) 5--16] showed that the minimum degree of a minimal matching covered graph different from $K_2$ is either 2 or 3. In this paper, we prove that every minimal matching covered graph with at least 4 vertices contains at least two nonadjacent edges, each of which is either a 2-line or a 3-line. Consequently, we show that every minimal matching covered graph with at least 4 vertices and minimum degree 3 contains at least 4 vertices of degree 3. Furthermore, the lower bounds for both the number of 3-lines and the number of cubic vertices are sharp.

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arXiv2025

On minimal k-factor-critical planar graphs

A graph of order $n$ is said to be \emph{$k$-factor-critical} ($0\leq k <n$) if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical for any edge $e$ in $G$. Favaron and Shi posed the conjecture that every minimal $k$-factor-critical graph is of minimum degree $k+1$ in 1998. In this paper, we confirm the conjecture for planar graphs.

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arXiv2025

The minimum degree of minimal 2-extendable claw-free graphs

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal if the deletion of any edge results in a graph that is not $k$-extendable. In 1994, Plummer proved that every $k$-extendable claw-free graph has minimum degree at least $2k$. Recently, He et al. showed that every minimal 1-extendable graph has minimum degree 2 or 3. In this paper, we prove that the minimum degree of a minimal 2-extendable claw-free graph is either $4$ or $5$.

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arXiv2025

Nice vertices in cubic graphs

A subgraph $G'$ of a graph $G$ is nice if $G-V(G')$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and its neighbors induce a nice subgraph. D. Kr\'{a}l et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph $G$ is 2-connected, then $G$ has at least 4 nice vertices; if $G$ is 3-connected and $G\neq K_4$, then $G$ has at least 6 nice vertices. We also determine all the corresponding extremal graphs. For a cubic bipartite graph $G$ with bipartition $(A,B)$, a pair of vertices $a\in A$ and $b\in B$ is called a nice pair if $a$ and $b$ together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph $G$ is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that $G$ has at least 9 nice pairs of vertices and $K_{3,3}$ is the only extremal graph.

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arXiv2024

Wheel-like bricks and minimal matching covered graphs

A connected graph G with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge e in a matching covered graph G is removable if G-e is matching covered. A pair {e; f} of edges of a matching covered graph G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A 3-connected graph is a brick if the removal of any two distinct vertices, the left graph has a perfect matching. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty proposed a problem of characterizing wheel-like bricks. We show that every wheel-like brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels in a certain manner. A matching covered graph is minimal if the removal of any edge, the left graph is not matching covered. Lovasz and Plummer proved that the minimum degree of a minimal matching covered bipartite graph different from K2 is 2 by ear decompositions in 1977. By the properties of wheel-like bricks, we prove that the minimum degree of a minimal matching covered graph other than K2 is 2 or 3.

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arXiv2024

Cubic bricks that every b-invariant edge is forcing

A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges. Denote by b(G) the number of bricks of G. An edge e of G is removable if G-e is also matching covered, and solitary (or forcing) if after the removal of the two end vertices of e, the left graph has a unique perfect matching. Furthermore, a removable edge e of a brick G is b-invariant if b(G-e) = 1. Lucchesi and Murty proposed a problem of characterizing bricks, distinct from K4, the prism and the Petersen graph, in which every b-invariant edge is forcing. We answer the problem for cubic bricks by showing that there are exactly ten cubic bricks, including K4, the prism and the Petersen graph, every b-invariant edge of which is forcing.

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arXiv2024

Planar wheel-like bricks

An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.

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arXiv2024

The number of perfect matchings in a brick

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovasz and Plummer. Lucchesi and Murty conjectured that there exists a positive integer N such that for every n>N, every brick on n vertices has at least n-1 perfect matchings. We present an infinite family of bricks such that for each even integer n (n > 17), there exists a brick with n vertices in this family that contains [0:625n] perfect matchings, showing that this conjecture fails.

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arXiv2024

Removable edges in near-bipartite bricks

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lov\'asz and Plummer. A nonbipartite matching covered graph $G$ is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi, and Murty proved that every brick other than $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1,e_2\}$ such that $G-\{e_1,e_2\}$ is a bipartite matching covered graph. In this paper, we show that in a near-bipartite brick $G$ with at least six vertices, every vertex of $G$, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, $G$ has at least $\frac{|V(G)|-6}{2}$ removable edges. Moreover, all graphs attaining this lower bound are characterized.

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arXiv2023

The minimum degree of minimal $k$-factor-critical claw-free graphs*

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of it results in a graph that is not $k$-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimal $k$-factor-critical graph has minimum degree $k+1$. In this paper, we confirm the conjecture for minimal $k$-factor-critical claw-free graphs. Moreover, we show that every minimal $k$-factor-critical claw-free graph $G$ has at least $\frac{k-1}{2k}|V(G)|$ vertices of degree $k+1$ in the case of $(k+1)$-connected, yielding further evidence for S. Norine and R. Thomas' conjecture on the minimum degree of minimal bricks when $k=2$.

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arXiv2023

Counting spanning trees of (1, N)-periodic graphs

Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic graph whose vertices can be partitioned into two sets $V_1=\{v\mid\sigma(v)=v\}$ and $V_2=\{v\mid\sigma^i(v)\neq v\ \mbox{for any}\ 1<i<N\}$, where $\sigma$ is an automorphism with order $N$ of $G$. The subgraph of $G$ induced by $V_1$ is called a fixed subgraph. Yan and Zhang [Enumeration of spanning trees of graphs with rotational symmetry, J. Comb. Theory Ser. A, 118(2011): 1270-1290] studied the enumeration of spanning trees of a special type of (1, $N$)-periodic graphs with $V_1=\emptyset$ for any non-trivial automorphism with order $N$. In this paper, we obtain a concise formula for the number of spanning trees of (1, $N$)-periodic graphs. Our result can reduce to Yan and Zhang's when $V_1$ is empty. As applications, we give a new closed formula for the spanning tree generating function of cobweb lattices, and obtain formulae for the number of spanning trees of circulant graphs $C_n(s_1,s_2,\ldots,s_k)$ and $K_2\bigvee C_n(s_1,s_2,\ldots,s_k)$.

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arXiv2020

Laminar Tight Cuts in Matching Covered Graphs

An edge cut $C$ of a graph $G$ is {\it tight} if $|C \cap M|=1$ for every perfect matching $M$ of $G$.~Barrier cuts and 2-separation cuts are called {\it ELP-cuts}, which are two important types of tight cuts in matching covered graphs.~Edmonds, Lovász and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut.~Carvalho, Lucchesi, and Murty made a stronger conjecture: given any nontrivial tight cut $C$ in a matching covered graph $G$, there exists a nontrivial ELP-cut $D$ in $G$ which does not cross $C$.~We confirm the conjecture in this paper.

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arXiv2020

$b$-invariant edges in essentially 4-edge-connected near-bipartite cubic bricks

A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is {\em essentially 4-edge-connected} if it does not contain nontrivial 3-cuts. A brick $G$ is {\em near-bipartite} if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.

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