A novel approach to determining chromatic number induced by labelings

Abstract: Given a simple graph $G=(V,E)$ of order $p$ and size $q$, a bijection $f : V\cup E \to {1, 2, \ldots, p+q}$ is a local total neighborhood antimagic labeling of $G$ if the induced vertex coloring has the property $f^+_{tn}(u) \ne f<sup>+_{tn}(v)$ for every two adjacent vertices $u$ and $v$ where $f^+_{tn}(u) = \sum (f(ux) + f(x))$ over every neighbor $x$ of $u$. The local total neighborhood antimagic chromatic number of $G$, denoted $χ_{ltna}(G)$ is the minimum number of distinct induced colors over all local total neighborhood antimagic labeling of $G$. In this paper, we determine the local total neighborhood antimagic chromatic number of the join of graphs with distinct parity orders.

• The paper develops a matrix-based labeling framework that controls induced vertex colors in joins by combining constant-column-sum arrays with arithmetic row-sum progressions.
• The paper proves exact values of 3 for joins involving matching graphs or certain even cycles with odd null graphs, and exact values of 5 for corresponding joins with even wheels or odd cycles.
• The results show that the local total neighborhood antimagic chromatic number equals the ordinary chromatic number for all studied families, while leaving even-even joins and general equality conditions open.

Background and motivation

The paper studies the local total neighborhood antimagic chromatic number, denoted $\chi_{ltna}(G)$, of a simple $(p,q)$-graph $G$. A bijective total labeling $f : V(G)\cup E(G) \to [1,p+q]$ is a local total neighborhood antimagic labeling if the induced vertex color

$f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$

differs for every pair of adjacent vertices. The parameter $\chi_{ltna}(G)$ is the minimum number of distinct induced colors over all such labelings, and it satisfies $\chi_{ltna}(G)\ge \chi(G)$ whenever a labeling exists. This notion extends both the classical antimagic conjecture of Hartsfield and Ringel and the local antimagic chromatic number introduced by Arumugam et al., as well as the total neighborhood antimagic variant initiated by Shrimali and Parmar; the immediate predecessor is the recent preprint of Gao, Lau, Shiu and Yow, which established general bounds and exact values for standard graphs.

Two standing conjectures frame the area: that every connected graph other than $K_2$ is antimagic, and that every graph admits a local total neighborhood antimagic labeling. The present paper does not address these conjectures directly; instead it develops constructive machinery for joins with components of distinct parity orders.

A matrix-based framework for joins

The central methodological contribution is a structured matrix construction. Suppose $G$ is a $(p,q)$-graph of even order $(p,q)$0 whose vertices can be partitioned into $(p,q)$1 partite sets $(p,q)$2 of sizes $(p,q)$3, and suppose $(p,q)$4 admits a total labeling $(p,q)$5 under which the induced colors within each $(p,q)$6 form an arithmetic sequence (or collapse to a single value when $(p,q)$7). Condition (A) then requires a $(p,q)$8 array $(p,q)$9 over $G$0 with constant column sum $G$1, whose row sums partition into $G$2 sets forming arithmetic sequences aligned with those induced by $G$3.

Under two non-collision inequalities — that $G$4 values are pairwise distinct across partite sets, and that no color on $G$5 collides with the common color attained on the second component — the authors prove that $G$6, with equality when $G$7. The proof assigns labels from $G$8 to $G$9, shifted entries of $f : V(G)\cup E(G) \to [1,p+q]$0 to cross edges, and consecutive large integers to the isolated vertices; the constant column sums force all vertices in each $f : V(G)\cup E(G) \to [1,p+q]$1 to share one color, while the row-sum arithmetic progressions control inter-partite distinctness.

Two explicit families of arrays are constructed for $f : V(G)\cup E(G) \to [1,p+q]$2 and odd $f : V(G)\cup E(G) \to [1,p+q]$3: the array $f : V(G)\cup E(G) \to [1,p+q]$4, built from paired blocks $f : V(G)\cup E(G) \to [1,p+q]$5 yielding row sums in arithmetic progressions with common difference $f : V(G)\cup E(G) \to [1,p+q]$6, and the array $f : V(G)\cup E(G) \to [1,p+q]$7, yielding common difference $f : V(G)\cup E(G) \to [1,p+q]$8. Both have constant column sum $f : V(G)\cup E(G) \to [1,p+q]$9, and the case $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$0 is verified separately to be consistent. These arrays satisfy Condition (A) and drive all subsequent applications.

Exact results for specific joins

Applying the framework, the paper establishes five exact values, each matching the trivial lower bound $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$1:

Graph	Value
$f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$2, $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$3	3
$f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$4, $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$5	3
$f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$6, $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$7, $f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$8	5
$f^+_{tn}(u) = \sum_{ux\in E}\big[f(ux)+f(x)\big]$9, $\chi_{ltna}(G)$0	5
$\chi_{ltna}(G)$1 and $\chi_{ltna}(G)$2	5

The first two extend known results for $\chi_{ltna}(G)$3 and $\chi_{ltna}(G)$4 to joins with arbitrary odd-order null graphs. Since $\chi_{ltna}(G)$5 and $\chi_{ltna}(G)$6 are bipartite, the value 3 is optimal. The verifications require extensive case analysis: for instance, in the proof for $\chi_{ltna}(G)$7, the inequality in condition (ii) reduces to quadratic forms in $\chi_{ltna}(G)$8 whose non-vanishing is certified by showing their discriminants are not perfect squares, together with sign arguments splitting the parameter range according to the relative sizes of $\chi_{ltna}(G)$9 and $\chi_{ltna}(G)\ge \chi(G)$0.

For the wheel and odd-cycle joins, the paper first proves a generalization, Theorem on $\chi_{ltna}(G)\ge \chi(G)$1: if $\chi_{ltna}(G)\ge \chi(G)$2 itself admits an optimal local total neighborhood antimagic $\chi_{ltna}(G)\ge \chi(G)$3-coloring $\chi_{ltna}(G)\ge \chi(G)$4 satisfying four compatibility conditions (non-collision of combined colors, preservation of equal-color classes under degree constraints, and avoidance of degree-scaled color differences), then $\chi_{ltna}(G)\ge \chi(G)$5, with equality when $\chi_{ltna}(G)\ge \chi(G)$6. A corollary simplifies this when $\chi_{ltna}(G)\ge \chi(G)$7 is regular. Since $\chi_{ltna}(G)\ge \chi(G)$8 and $\chi_{ltna}(G)\ge \chi(G)$9 admit 3-colorings, and $K_2$0 and $K_2$1 contribute 2 colors, the bound $K_2$2 matches the clique lower bound in each case. The resulting proofs again split into parity cases for $K_2$3 and subcases indexed by $K_2$4 or $K_2$5, with finitely many exceptional parameter pairs resolved by direct computation (for example, $K_2$6, $K_2$7 in the $K_2$8 analysis).

An implication worth noting: across all treated families, $K_2$9 exactly, suggesting that for joins of parity-mismatched components the antimagic constraint imposes no overhead beyond the ordinary chromatic requirement — though the paper proves this only for the listed families, not in general.

Limitations and open questions

The framework carries substantial hypotheses. Condition (A) requires an array with constant column sums and arithmetic row sums; the explicit constructions cover only even-order $G$0 joined with odd-order components, so even-even and odd-odd joins fall outside the scope of this paper. Moreover, the compatibility conditions (i)–(iv) in the general join theorem must be checked ad hoc for each pair $G$1, and the proofs rely on discriminant-based non-vanishing checks that do not generalize transparently. The paper also presupposes that $G$2 and $G$3 individually admit suitable labelings, inheriting dependence on the unproven conjecture that every graph admits a local total neighborhood antimagic labeling.

The authors explicitly pose four open problems: determining $G$4 for $G$5 joined with $G$6, $G$7, or $G$8 (the even-cycle case not covered here); finding necessary and sufficient conditions for $G$9; establishing tight upper and lower bounds for general joins; and clarifying the relationship between $(p,q)$0 and $(p,q)$1.

Conclusion

The paper contributes a reusable matrix construction that converts arithmetic structure in vertex labelings of one factor and row sums of a magic-type array into controlled induced colors on a join, yielding exact values $(p,q)$2 for $(p,q)$3 and $(p,q)$4 joined with odd null graphs, and $(p,q)$5 for the corresponding wheel and odd-cycle joins. Its reach is bounded by the parity restriction and the per-case verification burden, and the stated open problems identify the natural next targets: even cycles as the first factor, a characterization of when the chromatic lower bound is attained, and the connection to the local antimagic chromatic number.